Theorem dfoprab4f 6421

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 1581	. . . . 5 |- F/ x w = <. t , u >.
2		dfoprab4f.x	. . . . . 6 |- F/ x ph
3		nfs1v 1999	. . . . . 6 |- F/ x [ t / x ] [ u / y ] ps
4	2, 3	nfbi 1642	. . . . 5 |- F/ x ( ph <-> [ t / x ] [ u / y ] ps )
5	1, 4	nfim 1625	. . . 4 |- F/ x ( w = <. t , u >. -> ( ph <-> [ t / x ] [ u / y ] ps ) )
6		opeq1 3902	. . . . . 6 |- ( x = t -> <. x , u >. = <. t , u >. )
7	6	eqeq2d 2250	. . . . 5 |- ( x = t -> ( w = <. x , u >. <-> w = <. t , u >. ) )
8		sbequ12 1824	. . . . . 6 |- ( x = t -> ( [ u / y ] ps <-> [ t / x ] [ u / y ] ps ) )
9	8	bibi2d 232	. . . . 5 |- ( x = t -> ( ( ph <-> [ u / y ] ps ) <-> ( ph <-> [ t / x ] [ u / y ] ps ) ) )
10	7, 9	imbi12d 234	. . . 4 |- ( x = t -> ( ( w = <. x , u >. -> ( ph <-> [ u / y ] ps ) ) <-> ( w = <. t , u >. -> ( ph <-> [ t / x ] [ u / y ] ps ) ) ) )
11		nfv 1581	. . . . . 6 |- F/ y w = <. x , u >.
12		dfoprab4f.y	. . . . . . 7 |- F/ y ph
13		nfs1v 1999	. . . . . . 7 |- F/ y [ u / y ] ps
14	12, 13	nfbi 1642	. . . . . 6 |- F/ y ( ph <-> [ u / y ] ps )
15	11, 14	nfim 1625	. . . . 5 |- F/ y ( w = <. x , u >. -> ( ph <-> [ u / y ] ps ) )
16		opeq2 3903	. . . . . . 7 |- ( y = u -> <. x , y >. = <. x , u >. )
17	16	eqeq2d 2250	. . . . . 6 |- ( y = u -> ( w = <. x , y >. <-> w = <. x , u >. ) )
18		sbequ12 1824	. . . . . . 7 |- ( y = u -> ( ps <-> [ u / y ] ps ) )
19	18	bibi2d 232	. . . . . 6 |- ( y = u -> ( ( ph <-> ps ) <-> ( ph <-> [ u / y ] ps ) ) )
20	17, 19	imbi12d 234	. . . . 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 1810	. . . 4 |- ( w = <. x , u >. -> ( ph <-> [ u / y ] ps ) )
23	5, 10, 22	chvar 1810	. . 3 |- ( w = <. t , u >. -> ( ph <-> [ t / x ] [ u / y ] ps ) )
24	23	dfoprab4 6420	. 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 1581	. . 3 |- F/ t ( ( x e. A /\ y e. B ) /\ ps )
26		nfv 1581	. . 3 |- F/ u ( ( x e. A /\ y e. B ) /\ ps )
27		nfv 1581	. . . 4 |- F/ x ( t e. A /\ u e. B )
28	27, 3	nfan 1618	. . 3 |- F/ x ( ( t e. A /\ u e. B ) /\ [ t / x ] [ u / y ] ps )
29		nfv 1581	. . . 4 |- F/ y ( t e. A /\ u e. B )
30	13	nfsb 2006	. . . 4 |- F/ y [ t / x ] [ u / y ] ps
31	29, 30	nfan 1618	. . 3 |- F/ y ( ( t e. A /\ u e. B ) /\ [ t / x ] [ u / y ] ps )
32		eleq1 2301	. . . . 5 |- ( x = t -> ( x e. A <-> t e. A ) )
33		eleq1 2301	. . . . 5 |- ( y = u -> ( y e. B <-> u e. B ) )
34	32, 33	bi2anan9 614	. . . 4 |- ( ( x = t /\ y = u ) -> ( ( x e. A /\ y e. B ) <-> ( t e. A /\ u e. B ) ) )
35	18, 8	sylan9bbr 467	. . . 4 |- ( ( x = t /\ y = u ) -> ( ps <-> [ t / x ] [ u / y ] ps ) )
36	34, 35	anbi12d 477	. . 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 6156	. 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 2262	1 |- { <. w , z >. | ( w e. ( A X. B ) /\ ph ) } = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ps ) }

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   F/wnf 1513   [wsb 1815   e. wcel 2209   <.cop 3711   {copab 4189   X. cxp 4770   {coprab 6080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-fo 5381  df-fv 5383  df-oprab 6083  df-1st 6368  df-2nd 6369

This theorem is referenced by: (None)