Theorem abfii2 4577

Description: Two ways to express the collection of finite intersections of a set A.

Hypothesis

Ref	Expression
abfii2.1	|- A e. V

Assertion

Ref	Expression
abfii2	|- {x | E.y(y (_ A /\ y e. Fin /\ x = |^|y)} = |^|{x | A.y((y (_ A /\ y =/= (/) /\ y e. Fin) -> |^|y e. x)}

Distinct variable group:   x,y,A

Proof of Theorem abfii2

Step	Hyp	Ref	Expression
1		abfii2.1	. . . . 5 |- A e. V
2	1	uniex 2886	. . . 4 |- U.A e. V
3	2	inex2 2732	. . 3 |- (|^|y i^i U.A) e. V
4		df-sn 2424	. . . . . 6 |- {(|^|y i^i U.A)} = {x | x = (|^|y i^i U.A)}
5		snex 2766	. . . . . 6 |- {(|^|y i^i U.A)} e. V
6	4, 5	eqeltrri 1552	. . . . 5 |- {x | x = (|^|y i^i U.A)} e. V
7	1, 6	abexssex 3888	. . . 4 |- {x | E.y(y (_ A /\ x = (|^|y i^i U.A))} e. V
8		3simp1 792	. . . . . . 7 |- ((y (_ A /\ y =/= (/) /\ y e. Fin) -> y (_ A)
9	8	anim1i 334	. . . . . 6 |- (((y (_ A /\ y =/= (/) /\ y e. Fin) /\ x = (|^|y i^i U.A)) -> (y (_ A /\ x = (|^|y i^i U.A)))
10	9	19.22i 1044	. . . . 5 |- (E.y((y (_ A /\ y =/= (/) /\ y e. Fin) /\ x = (|^|y i^i U.A)) -> E.y(y (_ A /\ x = (|^|y i^i U.A)))
11	10	ss2abi 2131	. . . 4 |- {x | E.y((y (_ A /\ y =/= (/) /\ y e. Fin) /\ x = (|^|y i^i U.A))} (_ {x | E.y(y (_ A /\ x = (|^|y i^i U.A))}
12	7, 11	ssexi 2735	. . 3 |- {x | E.y((y (_ A /\ y =/= (/) /\ y e. Fin) /\ x = (|^|y i^i U.A))} e. V
13	3, 12	intab 2574	. 2 |- |^|{x | A.y((y (_ A /\ y =/= (/) /\ y e. Fin) -> (|^|y i^i U.A) e. x)} = {x | E.y((y (_ A /\ y =/= (/) /\ y e. Fin) /\ x = (|^|y i^i U.A))}
14		intssuni2 2570	. . . . . . . . 9 |- ((y (_ A /\ y =/= (/)) -> |^|y (_ U.A)
15		dfss 2065	. . . . . . . . 9 |- (|^|y (_ U.A <-> |^|y = (|^|y i^i U.A))
16	14, 15	sylib 198	. . . . . . . 8 |- ((y (_ A /\ y =/= (/)) -> |^|y = (|^|y i^i U.A))
17	16	3adant3 803	. . . . . . 7 |- ((y (_ A /\ y =/= (/) /\ y e. Fin) -> |^|y = (|^|y i^i U.A))
18	17	eleq1d 1547	. . . . . 6 |- ((y (_ A /\ y =/= (/) /\ y e. Fin) -> (|^|y e. x <-> (|^|y i^i U.A) e. x))
19	18	pm5.74i 587	. . . . 5 |- (((y (_ A /\ y =/= (/) /\ y e. Fin) -> |^|y e. x) <-> ((y (_ A /\ y =/= (/) /\ y e. Fin) -> (|^|y i^i U.A) e. x))
20	19	albii 1003	. . . 4 |- (A.y((y (_ A /\ y =/= (/) /\ y e. Fin) -> |^|y e. x) <-> A.y((y (_ A /\ y =/= (/) /\ y e. Fin) -> (|^|y i^i U.A) e. x))
21	20	abbii 1582	. . 3 |- {x | A.y((y (_ A /\ y =/= (/) /\ y e. Fin) -> |^|y e. x)} = {x | A.y((y (_ A /\ y =/= (/) /\ y e. Fin) -> (|^|y i^i U.A) e. x)}
22	21	inteqi 2551	. 2 |- |^|{x | A.y((y (_ A /\ y =/= (/) /\ y e. Fin) -> |^|y e. x)} = |^|{x | A.y((y (_ A /\ y =/= (/) /\ y e. Fin) -> (|^|y i^i U.A) e. x)}
23		df-3an 781	. . . . 5 |- ((y (_ A /\ y e. Fin /\ x = |^|y) <-> ((y (_ A /\ y e. Fin) /\ x = |^|y))
24		visset 1820	. . . . . . . . 9 |- x e. V
25		eleq1 1541	. . . . . . . . 9 |- (x = |^|y -> (x e. V <-> |^|y e. V))
26	24, 25	mpbii 193	. . . . . . . 8 |- (x = |^|y -> |^|y e. V)
27		intex 2744	. . . . . . . 8 |- (y =/= (/) <-> |^|y e. V)
28	26, 27	sylibr 200	. . . . . . 7 |- (x = |^|y -> y =/= (/))
29	28	pm4.71ri 641	. . . . . 6 |- (x = |^|y <-> (y =/= (/) /\ x = |^|y))
30	29	anbi2i 483	. . . . 5 |- (((y (_ A /\ y e. Fin) /\ x = |^|y) <-> ((y (_ A /\ y e. Fin) /\ (y =/= (/) /\ x = |^|y)))
31		an4 509	. . . . . 6 |- (((y (_ A /\ y e. Fin) /\ (y =/= (/) /\ x = |^|y)) <-> ((y (_ A /\ y =/= (/)) /\ (y e. Fin /\ x = |^|y)))
32		df-3an 781	. . . . . . . 8 |- ((y (_ A /\ y =/= (/) /\ y e. Fin) <-> ((y (_ A /\ y =/= (/)) /\ y e. Fin))
33	32	anbi1i 484	. . . . . . 7 |- (((y (_ A /\ y =/= (/) /\ y e. Fin) /\ x = |^|y) <-> (((y (_ A /\ y =/= (/)) /\ y e. Fin) /\ x = |^|y))
34	16	eqeq2d 1493	. . . . . . . . 9 |- ((y (_ A /\ y =/= (/)) -> (x = |^|y <-> x = (|^|y i^i U.A)))
35	34	3adant3 803	. . . . . . . 8 |- ((y (_ A /\ y =/= (/) /\ y e. Fin) -> (x = |^|y <-> x = (|^|y i^i U.A)))
36	35	pm5.32i 648	. . . . . . 7 |- (((y (_ A /\ y =/= (/) /\ y e. Fin) /\ x = |^|y) <-> ((y (_ A /\ y =/= (/) /\ y e. Fin) /\ x = (|^|y i^i U.A)))
37		anass 442	. . . . . . 7 |- ((((y (_ A /\ y =/= (/)) /\ y e. Fin) /\ x = |^|y) <-> ((y (_ A /\ y =/= (/)) /\ (y e. Fin /\ x = |^|y)))
38	33, 36, 37	3bitr3ri 182	. . . . . 6 |- (((y (_ A /\ y =/= (/)) /\ (y e. Fin /\ x = |^|y)) <-> ((y (_ A /\ y =/= (/) /\ y e. Fin) /\ x = (|^|y i^i U.A)))
39	31, 38	bitri 173	. . . . 5 |- (((y (_ A /\ y e. Fin) /\ (y =/= (/) /\ x = |^|y)) <-> ((y (_ A /\ y =/= (/) /\ y e. Fin) /\ x = (|^|y i^i U.A)))
40	23, 30, 39	3bitri 177	. . . 4 |- ((y (_ A /\ y e. Fin /\ x = |^|y) <-> ((y (_ A /\ y =/= (/) /\ y e. Fin) /\ x = (|^|y i^i U.A)))
41	40	exbii 1055	. . 3 |- (E.y(y (_ A /\ y e. Fin /\ x = |^|y) <-> E.y((y (_ A /\ y =/= (/) /\ y e. Fin) /\ x = (|^|y i^i U.A)))
42	41	abbii 1582	. 2 |- {x | E.y(y (_ A /\ y e. Fin /\ x = |^|y)} = {x | E.y((y (_ A /\ y =/= (/) /\ y e. Fin) /\ x = (|^|y i^i U.A))}
43	13, 22, 42	3eqtr4ri 1513	1 |- {x | E.y(y (_ A /\ y e. Fin /\ x = |^|y)} = |^|{x | A.y((y (_ A /\ y =/= (/) /\ y e. Fin) -> |^|y e. x)}

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 779  A.wal 958   = wceq 960   e. wcel 962  E.wex 984  {cab 1469   =/= wne 1592  Vcvv 1818   i^i cin 2057   (_ wss 2058  (/)c0 2291  {csn 2421  U.cuni 2517  |^|cint 2547  Fincfn 4385

This theorem is referenced by:  abfii3 4578  abfii4 4579  abfii5 4580

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-9 969  ax-10 970  ax-11 971  ax-12 972  ax-13 973  ax-14 974  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1127  ax-10o 1144  ax-16 1214  ax-11o 1222  ax-ext 1464  ax-rep 2708  ax-sep 2718  ax-nul 2725  ax-pow 2758  ax-pr 2795  ax-un 2882

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 781  df-ex 985  df-sb 1176  df-eu 1386  df-mo 1387  df-clab 1470  df-cleq 1475  df-clel 1478  df-ne 1594  df-ral 1656  df-rex 1657  df-rab 1659  df-v 1819  df-sbc 1949  df-csb 2012  df-dif 2060  df-un 2061  df-in 2062  df-ss 2064  df-nul 2292  df-pw 2414  df-sn 2424  df-pr 2425  df-op 2428  df-uni 2518  df-int 2548  df-iun 2582  df-br 2635  df-opab 2682  df-id 2851  df-xp 3200  df-rel 3201  df-cnv 3202  df-co 3203  df-dm 3204  df-rn 3205  df-res 3206  df-ima 3207  df-fun 3208  df-fv 3214

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