Theorem unielxp 4123

Description: The membership relation for a cross product is inherited by union.

Assertion

Ref	Expression
unielxp	|- (A e. (B X. C) -> U.A e. U.(B X. C))

Proof of Theorem unielxp

Step	Hyp	Ref	Expression
1		elxp7 4119	. 2 |- (A e. (B X. C) <-> (A e. (V X. V) /\ ((1st` A) e. B /\ (2nd` A) e. C)))
2		elvvuni 3245	. . . 4 |- (A e. (V X. V) -> U.A e. A)
3	2	adantr 391	. . 3 |- ((A e. (V X. V) /\ ((1st` A) e. B /\ (2nd` A) e. C)) -> U.A e. A)
4		simprl 416	. . . . . 6 |- ((U.A e. A /\ (A e. (V X. V) /\ ((1st` A) e. B /\ (2nd` A) e. C))) -> A e. (V X. V))
5		eleq2 1542	. . . . . . . 8 |- (x = A -> (U.A e. x <-> U.A e. A))
6		eleq1 1541	. . . . . . . . 9 |- (x = A -> (x e. (V X. V) <-> A e. (V X. V)))
7		fveq2 3740	. . . . . . . . . . 11 |- (x = A -> (1st` x) = (1st` A))
8	7	eleq1d 1547	. . . . . . . . . 10 |- (x = A -> ((1st` x) e. B <-> (1st` A) e. B))
9		fveq2 3740	. . . . . . . . . . 11 |- (x = A -> (2nd` x) = (2nd` A))
10	9	eleq1d 1547	. . . . . . . . . 10 |- (x = A -> ((2nd` x) e. C <-> (2nd` A) e. C))
11	8, 10	anbi12d 631	. . . . . . . . 9 |- (x = A -> (((1st` x) e. B /\ (2nd` x) e. C) <-> ((1st` A) e. B /\ (2nd` A) e. C)))
12	6, 11	anbi12d 631	. . . . . . . 8 |- (x = A -> ((x e. (V X. V) /\ ((1st` x) e. B /\ (2nd` x) e. C)) <-> (A e. (V X. V) /\ ((1st` A) e. B /\ (2nd` A) e. C))))
13	5, 12	anbi12d 631	. . . . . . 7 |- (x = A -> ((U.A e. x /\ (x e. (V X. V) /\ ((1st` x) e. B /\ (2nd` x) e. C))) <-> (U.A e. A /\ (A e. (V X. V) /\ ((1st` A) e. B /\ (2nd` A) e. C)))))
14	13	cla4egv 1870	. . . . . 6 |- (A e. (V X. V) -> ((U.A e. A /\ (A e. (V X. V) /\ ((1st` A) e. B /\ (2nd` A) e. C))) -> E.x(U.A e. x /\ (x e. (V X. V) /\ ((1st` x) e. B /\ (2nd` x) e. C)))))
15	4, 14	mpcom 49	. . . . 5 |- ((U.A e. A /\ (A e. (V X. V) /\ ((1st` A) e. B /\ (2nd` A) e. C))) -> E.x(U.A e. x /\ (x e. (V X. V) /\ ((1st` x) e. B /\ (2nd` x) e. C))))
16		eluniab 2527	. . . . 5 |- (U.A e. U.{x | (x e. (V X. V) /\ ((1st` x) e. B /\ (2nd` x) e. C))} <-> E.x(U.A e. x /\ (x e. (V X. V) /\ ((1st` x) e. B /\ (2nd` x) e. C))))
17	15, 16	sylibr 200	. . . 4 |- ((U.A e. A /\ (A e. (V X. V) /\ ((1st` A) e. B /\ (2nd` A) e. C))) -> U.A e. U.{x | (x e. (V X. V) /\ ((1st` x) e. B /\ (2nd` x) e. C))})
18		xp2 4121	. . . . . 6 |- (B X. C) = {x e. (V X. V) | ((1st` x) e. B /\ (2nd` x) e. C)}
19		df-rab 1659	. . . . . 6 |- {x e. (V X. V) | ((1st` x) e. B /\ (2nd` x) e. C)} = {x | (x e. (V X. V) /\ ((1st` x) e. B /\ (2nd` x) e. C))}
20	18, 19	eqtri 1502	. . . . 5 |- (B X. C) = {x | (x e. (V X. V) /\ ((1st` x) e. B /\ (2nd` x) e. C))}
21	20	unieqi 2525	. . . 4 |- U.(B X. C) = U.{x | (x e. (V X. V) /\ ((1st` x) e. B /\ (2nd` x) e. C))}
22	17, 21	syl6eleqr 1566	. . 3 |- ((U.A e. A /\ (A e. (V X. V) /\ ((1st` A) e. B /\ (2nd` A) e. C))) -> U.A e. U.(B X. C))
23	3, 22	mpancom 709	. 2 |- ((A e. (V X. V) /\ ((1st` A) e. B /\ (2nd` A) e. C)) -> U.A e. U.(B X. C))
24	1, 23	sylbi 199	1 |- (A e. (B X. C) -> U.A e. U.(B X. C))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 960   e. wcel 962  E.wex 984  {cab 1469  {crab 1655  Vcvv 1818  U.cuni 2517   X. cxp 3184  ` cfv 3198  1stc1st 4093  2ndc2nd 4094

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-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-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-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-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  df-1st 4095  df-2nd 4096

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