Theorem dfoprab4f 5850

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 1462	. . . . 5 |- F/ x w = <. t , u >.
2		dfoprab4f.x	. . . . . 6 |- F/ x ph
3		nfs1v 1857	. . . . . 6 |- F/ x [ t / x ] [ u / y ] ps
4	2, 3	nfbi 1522	. . . . 5 |- F/ x ( ph <-> [ t / x ] [ u / y ] ps )
5	1, 4	nfim 1505	. . . 4 |- F/ x ( w = <. t , u >. -> ( ph <-> [ t / x ] [ u / y ] ps ) )
6		opeq1 3578	. . . . . 6 |- ( x = t -> <. x , u >. = <. t , u >. )
7	6	eqeq2d 2093	. . . . 5 |- ( x = t -> ( w = <. x , u >. <-> w = <. t , u >. ) )
8		sbequ12 1695	. . . . . 6 |- ( x = t -> ( [ u / y ] ps <-> [ t / x ] [ u / y ] ps ) )
9	8	bibi2d 230	. . . . 5 |- ( x = t -> ( ( ph <-> [ u / y ] ps ) <-> ( ph <-> [ t / x ] [ u / y ] ps ) ) )
10	7, 9	imbi12d 232	. . . 4 |- ( x = t -> ( ( w = <. x , u >. -> ( ph <-> [ u / y ] ps ) ) <-> ( w = <. t , u >. -> ( ph <-> [ t / x ] [ u / y ] ps ) ) ) )
11		nfv 1462	. . . . . 6 |- F/ y w = <. x , u >.
12		dfoprab4f.y	. . . . . . 7 |- F/ y ph
13		nfs1v 1857	. . . . . . 7 |- F/ y [ u / y ] ps
14	12, 13	nfbi 1522	. . . . . 6 |- F/ y ( ph <-> [ u / y ] ps )
15	11, 14	nfim 1505	. . . . 5 |- F/ y ( w = <. x , u >. -> ( ph <-> [ u / y ] ps ) )
16		opeq2 3579	. . . . . . 7 |- ( y = u -> <. x , y >. = <. x , u >. )
17	16	eqeq2d 2093	. . . . . 6 |- ( y = u -> ( w = <. x , y >. <-> w = <. x , u >. ) )
18		sbequ12 1695	. . . . . . 7 |- ( y = u -> ( ps <-> [ u / y ] ps ) )
19	18	bibi2d 230	. . . . . 6 |- ( y = u -> ( ( ph <-> ps ) <-> ( ph <-> [ u / y ] ps ) ) )
20	17, 19	imbi12d 232	. . . . 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 1681	. . . 4 |- ( w = <. x , u >. -> ( ph <-> [ u / y ] ps ) )
23	5, 10, 22	chvar 1681	. . 3 |- ( w = <. t , u >. -> ( ph <-> [ t / x ] [ u / y ] ps ) )
24	23	dfoprab4 5849	. 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 1462	. . 3 |- F/ t ( ( x e. A /\ y e. B ) /\ ps )
26		nfv 1462	. . 3 |- F/ u ( ( x e. A /\ y e. B ) /\ ps )
27		nfv 1462	. . . 4 |- F/ x ( t e. A /\ u e. B )
28	27, 3	nfan 1498	. . 3 |- F/ x ( ( t e. A /\ u e. B ) /\ [ t / x ] [ u / y ] ps )
29		nfv 1462	. . . 4 |- F/ y ( t e. A /\ u e. B )
30	13	nfsb 1864	. . . 4 |- F/ y [ t / x ] [ u / y ] ps
31	29, 30	nfan 1498	. . 3 |- F/ y ( ( t e. A /\ u e. B ) /\ [ t / x ] [ u / y ] ps )
32		eleq1 2142	. . . . 5 |- ( x = t -> ( x e. A <-> t e. A ) )
33		eleq1 2142	. . . . 5 |- ( y = u -> ( y e. B <-> u e. B ) )
34	32, 33	bi2anan9 571	. . . 4 |- ( ( x = t /\ y = u ) -> ( ( x e. A /\ y e. B ) <-> ( t e. A /\ u e. B ) ) )
35	18, 8	sylan9bbr 451	. . . 4 |- ( ( x = t /\ y = u ) -> ( ps <-> [ t / x ] [ u / y ] ps ) )
36	34, 35	anbi12d 457	. . 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 5609	. 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 2105	1 |- { <. w , z >. | ( w e. ( A X. B ) /\ ph ) } = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ps ) }

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1285   F/wnf 1390   e. wcel 1434   [wsb 1686   <.cop 3409   {copab 3846   X. cxp 4369   {coprab 5544

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-fo 4938  df-fv 4940  df-oprab 5547  df-1st 5798  df-2nd 5799

This theorem is referenced by: (None)