Theorem dfoprab4f 6856

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 1763	. . . . 5 |- F/ x w = <. t , u >.
2		dfoprab4f.x	. . . . . 6 |- F/ x ph
3		nfs1v 2268	. . . . . 6 |- F/ x [ t / x ] [ u / y ] ps
4	2, 3	nfbi 2019	. . . . 5 |- F/ x ( ph <-> [ t / x ] [ u / y ] ps )
5	1, 4	nfim 2005	. . . 4 |- F/ x ( w = <. t , u >. -> ( ph <-> [ t / x ] [ u / y ] ps ) )
6		opeq1 4169	. . . . . 6 |- ( x = t -> <. x , u >. = <. t , u >. )
7	6	eqeq2d 2463	. . . . 5 |- ( x = t -> ( w = <. x , u >. <-> w = <. t , u >. ) )
8		sbequ12 2085	. . . . . 6 |- ( x = t -> ( [ u / y ] ps <-> [ t / x ] [ u / y ] ps ) )
9	8	bibi2d 320	. . . . 5 |- ( x = t -> ( ( ph <-> [ u / y ] ps ) <-> ( ph <-> [ t / x ] [ u / y ] ps ) ) )
10	7, 9	imbi12d 322	. . . 4 |- ( x = t -> ( ( w = <. x , u >. -> ( ph <-> [ u / y ] ps ) ) <-> ( w = <. t , u >. -> ( ph <-> [ t / x ] [ u / y ] ps ) ) ) )
11		nfv 1763	. . . . . 6 |- F/ y w = <. x , u >.
12		dfoprab4f.y	. . . . . . 7 |- F/ y ph
13		nfs1v 2268	. . . . . . 7 |- F/ y [ u / y ] ps
14	12, 13	nfbi 2019	. . . . . 6 |- F/ y ( ph <-> [ u / y ] ps )
15	11, 14	nfim 2005	. . . . 5 |- F/ y ( w = <. x , u >. -> ( ph <-> [ u / y ] ps ) )
16		opeq2 4170	. . . . . . 7 |- ( y = u -> <. x , y >. = <. x , u >. )
17	16	eqeq2d 2463	. . . . . 6 |- ( y = u -> ( w = <. x , y >. <-> w = <. x , u >. ) )
18		sbequ12 2085	. . . . . . 7 |- ( y = u -> ( ps <-> [ u / y ] ps ) )
19	18	bibi2d 320	. . . . . 6 |- ( y = u -> ( ( ph <-> ps ) <-> ( ph <-> [ u / y ] ps ) ) )
20	17, 19	imbi12d 322	. . . . 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 2108	. . . 4 |- ( w = <. x , u >. -> ( ph <-> [ u / y ] ps ) )
23	5, 10, 22	chvar 2108	. . 3 |- ( w = <. t , u >. -> ( ph <-> [ t / x ] [ u / y ] ps ) )
24	23	dfoprab4 6855	. 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 1763	. . 3 |- F/ t ( ( x e. A /\ y e. B ) /\ ps )
26		nfv 1763	. . 3 |- F/ u ( ( x e. A /\ y e. B ) /\ ps )
27		nfv 1763	. . . 4 |- F/ x ( t e. A /\ u e. B )
28	27, 3	nfan 2013	. . 3 |- F/ x ( ( t e. A /\ u e. B ) /\ [ t / x ] [ u / y ] ps )
29		nfv 1763	. . . 4 |- F/ y ( t e. A /\ u e. B )
30	13	nfsb 2271	. . . 4 |- F/ y [ t / x ] [ u / y ] ps
31	29, 30	nfan 2013	. . 3 |- F/ y ( ( t e. A /\ u e. B ) /\ [ t / x ] [ u / y ] ps )
32		eleq1 2519	. . . . 5 |- ( x = t -> ( x e. A <-> t e. A ) )
33		eleq1 2519	. . . . 5 |- ( y = u -> ( y e. B <-> u e. B ) )
34	32, 33	bi2anan9 885	. . . 4 |- ( ( x = t /\ y = u ) -> ( ( x e. A /\ y e. B ) <-> ( t e. A /\ u e. B ) ) )
35	18, 8	sylan9bbr 708	. . . 4 |- ( ( x = t /\ y = u ) -> ( ps <-> [ t / x ] [ u / y ] ps ) )
36	34, 35	anbi12d 718	. . 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 2478	1 |- { <. w , z >. | ( w e. ( A X. B ) /\ ph ) } = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ps ) }

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1446   F/wnf 1669   [wsb 1799   e. wcel 1889   <.cop 3976   {copab 4463   X. cxp 4835   {coprab 6296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-iota 5549  df-fun 5587  df-fv 5593  df-oprab 6299  df-1st 6798  df-2nd 6799

This theorem is referenced by: (None)