Theorem dfoprab4 6416

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 4878	. . . . . 6 |- ( A X. B ) C_ ( _V X. _V )
2	1	sseli 3244	. . . . 5 |- ( w e. ( A X. B ) -> w e. ( _V X. _V ) )
3	2	adantr 276	. . . 4 |- ( ( w e. ( A X. B ) /\ ph ) -> w e. ( _V X. _V ) )
4	3	pm4.71ri 396	. . 3 |- ( ( w e. ( A X. B ) /\ ph ) <-> ( w e. ( _V X. _V ) /\ ( w e. ( A X. B ) /\ ph ) ) )
5	4	opabbii 4193	. 2 |- { <. w , z >. | ( w e. ( A X. B ) /\ ph ) } = { <. w , z >. | ( w e. ( _V X. _V ) /\ ( w e. ( A X. B ) /\ ph ) ) }
6		eleq1 2301	. . . . 5 |- ( w = <. x , y >. -> ( w e. ( A X. B ) <-> <. x , y >. e. ( A X. B ) ) )
7		opelxp 4799	. . . . 5 |- ( <. x , y >. e. ( A X. B ) <-> ( x e. A /\ y e. B ) )
8	6, 7	bitrdi 196	. . . 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 477	. . 3 |- ( w = <. x , y >. -> ( ( w e. ( A X. B ) /\ ph ) <-> ( ( x e. A /\ y e. B ) /\ ps ) ) )
11	10	dfoprab3 6415	. 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 2259	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   e. wcel 2209   _Vcvv 2821   <.cop 3708   {copab 4186   X. cxp 4767   {coprab 6076

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 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fo 5378  df-fv 5380  df-oprab 6079  df-1st 6364  df-2nd 6365

This theorem is referenced by:  dfoprab4f  6417  dfxp3  6420