Theorem dfopab2 4129

Description: A way to define an ordered-pair class abstraction without using existential quantifiers.

Assertion

Ref	Expression
dfopab2	|- {<.x, y>. | ph} = {z | (z e. (V X. V) /\ [(1st` z) / x][(2nd` z) / y]ph)}

Distinct variable groups:   ph,z   x,y,z

Proof of Theorem dfopab2

Step	Hyp	Ref	Expression
1		df-opab 2682	. 2 |- {<.x, y>. | ph} = {z | E.xE.y(z = <.x, y>. /\ ph)}
2		fvex 3748	. . . . . 6 |- (1st` z) e. V
3	2	hbsbc1v 1957	. . . . 5 |- ([(1st` z) / x][(2nd` z) / y]ph -> A.x[(1st` z) / x][(2nd` z) / y]ph)
4	3	19.41 1099	. . . 4 |- (E.x(E.y z = <.x, y>. /\ [(1st` z) / x][(2nd` z) / y]ph) <-> (E.xE.y z = <.x, y>. /\ [(1st` z) / x][(2nd` z) / y]ph))
5		fveq2 3740	. . . . . . . . . . 11 |- (z = <.x, y>. -> (2nd` z) = (2nd` <.x, y>.))
6		visset 1820	. . . . . . . . . . . 12 |- x e. V
7		visset 1820	. . . . . . . . . . . 12 |- y e. V
8	6, 7	op2nd 4102	. . . . . . . . . . 11 |- (2nd` <.x, y>.) = y
9	5, 8	syl6req 1531	. . . . . . . . . 10 |- (z = <.x, y>. -> y = (2nd` z))
10		sbceq1a 1951	. . . . . . . . . 10 |- (y = (2nd` z) -> (ph <-> [(2nd` z) / y]ph))
11	9, 10	syl 10	. . . . . . . . 9 |- (z = <.x, y>. -> (ph <-> [(2nd` z) / y]ph))
12		fveq2 3740	. . . . . . . . . . 11 |- (z = <.x, y>. -> (1st` z) = (1st` <.x, y>.))
13	6	op1st 4101	. . . . . . . . . . 11 |- (1st` <.x, y>.) = x
14	12, 13	syl6req 1531	. . . . . . . . . 10 |- (z = <.x, y>. -> x = (1st` z))
15		sbceq1a 1951	. . . . . . . . . 10 |- (x = (1st` z) -> ([(2nd` z) / y]ph <-> [(1st` z) / x][(2nd` z) / y]ph))
16	14, 15	syl 10	. . . . . . . . 9 |- (z = <.x, y>. -> ([(2nd` z) / y]ph <-> [(1st` z) / x][(2nd` z) / y]ph))
17	11, 16	bitrd 531	. . . . . . . 8 |- (z = <.x, y>. -> (ph <-> [(1st` z) / x][(2nd` z) / y]ph))
18	17	pm5.32i 648	. . . . . . 7 |- ((z = <.x, y>. /\ ph) <-> (z = <.x, y>. /\ [(1st` z) / x][(2nd` z) / y]ph))
19	18	exbii 1055	. . . . . 6 |- (E.y(z = <.x, y>. /\ ph) <-> E.y(z = <.x, y>. /\ [(1st` z) / x][(2nd` z) / y]ph))
20		ax-17 975	. . . . . . . . 9 |- (x e. (1st` z) -> A.y x e. (1st` z))
21		fvex 3748	. . . . . . . . . 10 |- (2nd` z) e. V
22	21	hbsbc1v 1957	. . . . . . . . 9 |- ([(2nd` z) / y]ph -> A.y[(2nd` z) / y]ph)
23	20, 22	hbsbcg 1958	. . . . . . . 8 |- ((1st` z) e. V -> ([(1st` z) / x][(2nd` z) / y]ph -> A.y[(1st` z) / x][(2nd` z) / y]ph))
24	2, 23	ax-mp 7	. . . . . . 7 |- ([(1st` z) / x][(2nd` z) / y]ph -> A.y[(1st` z) / x][(2nd` z) / y]ph)
25	24	19.41 1099	. . . . . 6 |- (E.y(z = <.x, y>. /\ [(1st` z) / x][(2nd` z) / y]ph) <-> (E.y z = <.x, y>. /\ [(1st` z) / x][(2nd` z) / y]ph))
26	19, 25	bitri 173	. . . . 5 |- (E.y(z = <.x, y>. /\ ph) <-> (E.y z = <.x, y>. /\ [(1st` z) / x][(2nd` z) / y]ph))
27	26	exbii 1055	. . . 4 |- (E.xE.y(z = <.x, y>. /\ ph) <-> E.x(E.y z = <.x, y>. /\ [(1st` z) / x][(2nd` z) / y]ph))
28		elvv 3244	. . . . 5 |- (z e. (V X. V) <-> E.xE.y z = <.x, y>.)
29	28	anbi1i 484	. . . 4 |- ((z e. (V X. V) /\ [(1st` z) / x][(2nd` z) / y]ph) <-> (E.xE.y z = <.x, y>. /\ [(1st` z) / x][(2nd` z) / y]ph))
30	4, 27, 29	3bitr4i 183	. . 3 |- (E.xE.y(z = <.x, y>. /\ ph) <-> (z e. (V X. V) /\ [(1st` z) / x][(2nd` z) / y]ph))
31	30	abbii 1582	. 2 |- {z | E.xE.y(z = <.x, y>. /\ ph)} = {z | (z e. (V X. V) /\ [(1st` z) / x][(2nd` z) / y]ph)}
32	1, 31	eqtri 1502	1 |- {<.x, y>. | ph} = {z | (z e. (V X. V) /\ [(1st` z) / x][(2nd` z) / y]ph)}

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 958   = wceq 960   e. wcel 962  E.wex 984  [wsbc 1174  {cab 1469  Vcvv 1818  <.cop 2423  {copab 2681   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-v 1819  df-sbc 1949  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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