Theorem dfoprab4 6399

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 4863	. . . . . 6 |- ( A X. B ) C_ ( _V X. _V )
2	1	sseli 3238	. . . . 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 392	. . 3 |- ( ( w e. ( A X. B ) /\ ph ) <-> ( w e. ( _V X. _V ) /\ ( w e. ( A X. B ) /\ ph ) ) )
5	4	opabbii 4182	. 2 |- { <. w , z >. | ( w e. ( A X. B ) /\ ph ) } = { <. w , z >. | ( w e. ( _V X. _V ) /\ ( w e. ( A X. B ) /\ ph ) ) }
6		eleq1 2297	. . . . 5 |- ( w = <. x , y >. -> ( w e. ( A X. B ) <-> <. x , y >. e. ( A X. B ) ) )
7		opelxp 4784	. . . . 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 473	. . 3 |- ( w = <. x , y >. -> ( ( w e. ( A X. B ) /\ ph ) <-> ( ( x e. A /\ y e. B ) /\ ps ) ) )
11	10	dfoprab3 6398	. 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 2255	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 1398   e. wcel 2205   _Vcvv 2815   <.cop 3697   {copab 4175   X. cxp 4752   {coprab 6059

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fo 5363  df-fv 5365  df-oprab 6062  df-1st 6347  df-2nd 6348

This theorem is referenced by:  dfoprab4f  6400  dfxp3  6403