Theorem dfoprab4f 6091

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 1508	. . . . 5 |- F/ x w = <. t , u >.
2		dfoprab4f.x	. . . . . 6 |- F/ x ph
3		nfs1v 1912	. . . . . 6 |- F/ x [ t / x ] [ u / y ] ps
4	2, 3	nfbi 1568	. . . . 5 |- F/ x ( ph <-> [ t / x ] [ u / y ] ps )
5	1, 4	nfim 1551	. . . 4 |- F/ x ( w = <. t , u >. -> ( ph <-> [ t / x ] [ u / y ] ps ) )
6		opeq1 3705	. . . . . 6 |- ( x = t -> <. x , u >. = <. t , u >. )
7	6	eqeq2d 2151	. . . . 5 |- ( x = t -> ( w = <. x , u >. <-> w = <. t , u >. ) )
8		sbequ12 1744	. . . . . 6 |- ( x = t -> ( [ u / y ] ps <-> [ t / x ] [ u / y ] ps ) )
9	8	bibi2d 231	. . . . 5 |- ( x = t -> ( ( ph <-> [ u / y ] ps ) <-> ( ph <-> [ t / x ] [ u / y ] ps ) ) )
10	7, 9	imbi12d 233	. . . 4 |- ( x = t -> ( ( w = <. x , u >. -> ( ph <-> [ u / y ] ps ) ) <-> ( w = <. t , u >. -> ( ph <-> [ t / x ] [ u / y ] ps ) ) ) )
11		nfv 1508	. . . . . 6 |- F/ y w = <. x , u >.
12		dfoprab4f.y	. . . . . . 7 |- F/ y ph
13		nfs1v 1912	. . . . . . 7 |- F/ y [ u / y ] ps
14	12, 13	nfbi 1568	. . . . . 6 |- F/ y ( ph <-> [ u / y ] ps )
15	11, 14	nfim 1551	. . . . 5 |- F/ y ( w = <. x , u >. -> ( ph <-> [ u / y ] ps ) )
16		opeq2 3706	. . . . . . 7 |- ( y = u -> <. x , y >. = <. x , u >. )
17	16	eqeq2d 2151	. . . . . 6 |- ( y = u -> ( w = <. x , y >. <-> w = <. x , u >. ) )
18		sbequ12 1744	. . . . . . 7 |- ( y = u -> ( ps <-> [ u / y ] ps ) )
19	18	bibi2d 231	. . . . . 6 |- ( y = u -> ( ( ph <-> ps ) <-> ( ph <-> [ u / y ] ps ) ) )
20	17, 19	imbi12d 233	. . . . 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 1730	. . . 4 |- ( w = <. x , u >. -> ( ph <-> [ u / y ] ps ) )
23	5, 10, 22	chvar 1730	. . 3 |- ( w = <. t , u >. -> ( ph <-> [ t / x ] [ u / y ] ps ) )
24	23	dfoprab4 6090	. 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 1508	. . 3 |- F/ t ( ( x e. A /\ y e. B ) /\ ps )
26		nfv 1508	. . 3 |- F/ u ( ( x e. A /\ y e. B ) /\ ps )
27		nfv 1508	. . . 4 |- F/ x ( t e. A /\ u e. B )
28	27, 3	nfan 1544	. . 3 |- F/ x ( ( t e. A /\ u e. B ) /\ [ t / x ] [ u / y ] ps )
29		nfv 1508	. . . 4 |- F/ y ( t e. A /\ u e. B )
30	13	nfsb 1919	. . . 4 |- F/ y [ t / x ] [ u / y ] ps
31	29, 30	nfan 1544	. . 3 |- F/ y ( ( t e. A /\ u e. B ) /\ [ t / x ] [ u / y ] ps )
32		eleq1 2202	. . . . 5 |- ( x = t -> ( x e. A <-> t e. A ) )
33		eleq1 2202	. . . . 5 |- ( y = u -> ( y e. B <-> u e. B ) )
34	32, 33	bi2anan9 595	. . . 4 |- ( ( x = t /\ y = u ) -> ( ( x e. A /\ y e. B ) <-> ( t e. A /\ u e. B ) ) )
35	18, 8	sylan9bbr 458	. . . 4 |- ( ( x = t /\ y = u ) -> ( ps <-> [ t / x ] [ u / y ] ps ) )
36	34, 35	anbi12d 464	. . 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 5845	. 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 2163	1 |- { <. w , z >. | ( w e. ( A X. B ) /\ ph ) } = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ps ) }

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   F/wnf 1436   e. wcel 1480   [wsb 1735   <.cop 3530   {copab 3988   X. cxp 4537   {coprab 5775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fo 5129  df-fv 5131  df-oprab 5778  df-1st 6038  df-2nd 6039

This theorem is referenced by: (None)