Theorem dfoprab4 6098

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 4655	. . . . . 6 |- ( A X. B ) C_ ( _V X. _V )
2	1	sseli 3098	. . . . 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 4003	. 2 |- { <. w , z >. | ( w e. ( A X. B ) /\ ph ) } = { <. w , z >. | ( w e. ( _V X. _V ) /\ ( w e. ( A X. B ) /\ ph ) ) }
6		eleq1 2203	. . . . 5 |- ( w = <. x , y >. -> ( w e. ( A X. B ) <-> <. x , y >. e. ( A X. B ) ) )
7		opelxp 4577	. . . . 5 |- ( <. x , y >. e. ( A X. B ) <-> ( x e. A /\ y e. B ) )
8	6, 7	syl6bb 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 6097	. 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 2161	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 1332   e. wcel 1481   _Vcvv 2689   <.cop 3535   {copab 3996   X. cxp 4545   {coprab 5783

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-fo 5137  df-fv 5139  df-oprab 5786  df-1st 6046  df-2nd 6047

This theorem is referenced by:  dfoprab4f  6099  dfxp3  6100