Theorem dfoprab4 6160

Description: Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypothesis

Ref	Expression
dfoprab4.1	|- ( w = <. x , y >. -> ( ph <-> ps ) )

Assertion

Ref	Expression
dfoprab4	|- { <. w , z >. | ( w e. ( A X. B ) /\ ph ) } = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ps ) }

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

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

Proof of Theorem dfoprab4

Step	Hyp	Ref	Expression
1		xpss 4712	. . . . . 6 |- ( A X. B ) C_ ( _V X. _V )
2	1	sseli 3138	. . . . 5 |- ( w e. ( A X. B ) -> w e. ( _V X. _V ) )
3	2	adantr 274	. . . 4 |- ( ( w e. ( A X. B ) /\ ph ) -> w e. ( _V X. _V ) )
4	3	pm4.71ri 390	. . 3 |- ( ( w e. ( A X. B ) /\ ph ) <-> ( w e. ( _V X. _V ) /\ ( w e. ( A X. B ) /\ ph ) ) )
5	4	opabbii 4049	. 2 |- { <. w , z >. | ( w e. ( A X. B ) /\ ph ) } = { <. w , z >. | ( w e. ( _V X. _V ) /\ ( w e. ( A X. B ) /\ ph ) ) }
6		eleq1 2229	. . . . 5 |- ( w = <. x , y >. -> ( w e. ( A X. B ) <-> <. x , y >. e. ( A X. B ) ) )
7		opelxp 4634	. . . . 5 |- ( <. x , y >. e. ( A X. B ) <-> ( x e. A /\ y e. B ) )
8	6, 7	bitrdi 195	. . . 4 |- ( w = <. x , y >. -> ( w e. ( A X. B ) <-> ( x e. A /\ y e. B ) ) )
9		dfoprab4.1	. . . 4 |- ( w = <. x , y >. -> ( ph <-> ps ) )
10	8, 9	anbi12d 465	. . 3 |- ( w = <. x , y >. -> ( ( w e. ( A X. B ) /\ ph ) <-> ( ( x e. A /\ y e. B ) /\ ps ) ) )
11	10	dfoprab3 6159	. 2 |- { <. w , z >. | ( w e. ( _V X. _V ) /\ ( w e. ( A X. B ) /\ ph ) ) } = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ps ) }
12	5, 11	eqtri 2186	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 1343   e. wcel 2136   _Vcvv 2726   <.cop 3579   {copab 4042   X. cxp 4602   {coprab 5843

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fo 5194  df-fv 5196  df-oprab 5846  df-1st 6108  df-2nd 6109

This theorem is referenced by:  dfoprab4f  6161  dfxp3  6162