Theorem dfoprab4f 6857

Description: Operation class abstraction expressed without existential quantifiers. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypotheses

Ref	Expression
dfoprab4f.x	|- F/ x ph
dfoprab4f.y	|- F/ y ph
dfoprab4f.1	|- ( w = <. x , y >. -> ( ph <-> ps ) )

Assertion

Ref	Expression
dfoprab4f	|- { <. w , z >. | ( w e. ( A X. B ) /\ ph ) } = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ps ) }

Distinct variable groups:   x, w, y, z   w, A, x, y   w, B, x, y   ps, w

Allowed substitution hints:   ph( x, y, z, w)   ps( x, y, z)   A( z)   B( z)

Proof of Theorem dfoprab4f

Dummy variables u t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1708	. . . . 5 |- F/ x w = <. t , u >.
2		dfoprab4f.x	. . . . . 6 |- F/ x ph
3		nfs1v 2182	. . . . . 6 |- F/ x [ t / x ] [ u / y ] ps
4	2, 3	nfbi 1935	. . . . 5 |- F/ x ( ph <-> [ t / x ] [ u / y ] ps )
5	1, 4	nfim 1921	. . . 4 |- F/ x ( w = <. t , u >. -> ( ph <-> [ t / x ] [ u / y ] ps ) )
6		opeq1 4219	. . . . . 6 |- ( x = t -> <. x , u >. = <. t , u >. )
7	6	eqeq2d 2471	. . . . 5 |- ( x = t -> ( w = <. x , u >. <-> w = <. t , u >. ) )
8		sbequ12 1993	. . . . . 6 |- ( x = t -> ( [ u / y ] ps <-> [ t / x ] [ u / y ] ps ) )
9	8	bibi2d 318	. . . . 5 |- ( x = t -> ( ( ph <-> [ u / y ] ps ) <-> ( ph <-> [ t / x ] [ u / y ] ps ) ) )
10	7, 9	imbi12d 320	. . . 4 |- ( x = t -> ( ( w = <. x , u >. -> ( ph <-> [ u / y ] ps ) ) <-> ( w = <. t , u >. -> ( ph <-> [ t / x ] [ u / y ] ps ) ) ) )
11		nfv 1708	. . . . . 6 |- F/ y w = <. x , u >.
12		dfoprab4f.y	. . . . . . 7 |- F/ y ph
13		nfs1v 2182	. . . . . . 7 |- F/ y [ u / y ] ps
14	12, 13	nfbi 1935	. . . . . 6 |- F/ y ( ph <-> [ u / y ] ps )
15	11, 14	nfim 1921	. . . . 5 |- F/ y ( w = <. x , u >. -> ( ph <-> [ u / y ] ps ) )
16		opeq2 4220	. . . . . . 7 |- ( y = u -> <. x , y >. = <. x , u >. )
17	16	eqeq2d 2471	. . . . . 6 |- ( y = u -> ( w = <. x , y >. <-> w = <. x , u >. ) )
18		sbequ12 1993	. . . . . . 7 |- ( y = u -> ( ps <-> [ u / y ] ps ) )
19	18	bibi2d 318	. . . . . 6 |- ( y = u -> ( ( ph <-> ps ) <-> ( ph <-> [ u / y ] ps ) ) )
20	17, 19	imbi12d 320	. . . . 5 |- ( y = u -> ( ( w = <. x , y >. -> ( ph <-> ps ) ) <-> ( w = <. x , u >. -> ( ph <-> [ u / y ] ps ) ) ) )
21		dfoprab4f.1	. . . . 5 |- ( w = <. x , y >. -> ( ph <-> ps ) )
22	15, 20, 21	chvar 2014	. . . 4 |- ( w = <. x , u >. -> ( ph <-> [ u / y ] ps ) )
23	5, 10, 22	chvar 2014	. . 3 |- ( w = <. t , u >. -> ( ph <-> [ t / x ] [ u / y ] ps ) )
24	23	dfoprab4 6856	. 2 |- { <. w , z >. | ( w e. ( A X. B ) /\ ph ) } = { <. <. t , u >. , z >. | ( ( t e. A /\ u e. B ) /\ [ t / x ] [ u / y ] ps ) }
25		nfv 1708	. . 3 |- F/ t ( ( x e. A /\ y e. B ) /\ ps )
26		nfv 1708	. . 3 |- F/ u ( ( x e. A /\ y e. B ) /\ ps )
27		nfv 1708	. . . 4 |- F/ x ( t e. A /\ u e. B )
28	27, 3	nfan 1929	. . 3 |- F/ x ( ( t e. A /\ u e. B ) /\ [ t / x ] [ u / y ] ps )
29		nfv 1708	. . . 4 |- F/ y ( t e. A /\ u e. B )
30	13	nfsb 2185	. . . 4 |- F/ y [ t / x ] [ u / y ] ps
31	29, 30	nfan 1929	. . 3 |- F/ y ( ( t e. A /\ u e. B ) /\ [ t / x ] [ u / y ] ps )
32		eleq1 2529	. . . . 5 |- ( x = t -> ( x e. A <-> t e. A ) )
33		eleq1 2529	. . . . 5 |- ( y = u -> ( y e. B <-> u e. B ) )
34	32, 33	bi2anan9 873	. . . 4 |- ( ( x = t /\ y = u ) -> ( ( x e. A /\ y e. B ) <-> ( t e. A /\ u e. B ) ) )
35	18, 8	sylan9bbr 700	. . . 4 |- ( ( x = t /\ y = u ) -> ( ps <-> [ t / x ] [ u / y ] ps ) )
36	34, 35	anbi12d 710	. . 3 |- ( ( x = t /\ y = u ) -> ( ( ( x e. A /\ y e. B ) /\ ps ) <-> ( ( t e. A /\ u e. B ) /\ [ t / x ] [ u / y ] ps ) ) )
37	25, 26, 28, 31, 36	cbvoprab12 6370	. 2 |- { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ps ) } = { <. <. t , u >. , z >. | ( ( t e. A /\ u e. B ) /\ [ t / x ] [ u / y ] ps ) }
38	24, 37	eqtr4i 2489	1 |- { <. w , z >. | ( w e. ( A X. B ) /\ ph ) } = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ps ) }

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1395   F/wnf 1617   [wsb 1740   e. wcel 1819   <.cop 4038   {copab 4514   X. cxp 5006   {coprab 6297

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-iota 5557  df-fun 5596  df-fv 5602  df-oprab 6300  df-1st 6799  df-2nd 6800

This theorem is referenced by: (None)