Theorem resoprab 6151

Description: Restriction of an operation class abstraction. (Contributed by NM, 10-Feb-2007.)

Assertion

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

Distinct variable groups:   x, y, z, A   x, B, y, z

Allowed substitution hints:   ph( x, y, z)

Proof of Theorem resoprab

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		resopab 5084	. . 3 |- ( { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) } |` ( A X. B ) ) = { <. w , z >. | ( w e. ( A X. B ) /\ E. x E. y ( w = <. x , y >. /\ ph ) ) }
2		19.42vv 1963	. . . . 5 |- ( E. x E. y ( w e. ( A X. B ) /\ ( w = <. x , y >. /\ ph ) ) <-> ( w e. ( A X. B ) /\ E. x E. y ( w = <. x , y >. /\ ph ) ) )
3		an12 563	. . . . . . 7 |- ( ( w e. ( A X. B ) /\ ( w = <. x , y >. /\ ph ) ) <-> ( w = <. x , y >. /\ ( w e. ( A X. B ) /\ ph ) ) )
4		eleq1 2297	. . . . . . . . . 10 |- ( w = <. x , y >. -> ( w e. ( A X. B ) <-> <. x , y >. e. ( A X. B ) ) )
5		opelxp 4781	. . . . . . . . . 10 |- ( <. x , y >. e. ( A X. B ) <-> ( x e. A /\ y e. B ) )
6	4, 5	bitrdi 196	. . . . . . . . 9 |- ( w = <. x , y >. -> ( w e. ( A X. B ) <-> ( x e. A /\ y e. B ) ) )
7	6	anbi1d 465	. . . . . . . 8 |- ( w = <. x , y >. -> ( ( w e. ( A X. B ) /\ ph ) <-> ( ( x e. A /\ y e. B ) /\ ph ) ) )
8	7	pm5.32i 454	. . . . . . 7 |- ( ( w = <. x , y >. /\ ( w e. ( A X. B ) /\ ph ) ) <-> ( w = <. x , y >. /\ ( ( x e. A /\ y e. B ) /\ ph ) ) )
9	3, 8	bitri 184	. . . . . 6 |- ( ( w e. ( A X. B ) /\ ( w = <. x , y >. /\ ph ) ) <-> ( w = <. x , y >. /\ ( ( x e. A /\ y e. B ) /\ ph ) ) )
10	9	2exbii 1655	. . . . 5 |- ( E. x E. y ( w e. ( A X. B ) /\ ( w = <. x , y >. /\ ph ) ) <-> E. x E. y ( w = <. x , y >. /\ ( ( x e. A /\ y e. B ) /\ ph ) ) )
11	2, 10	bitr3i 186	. . . 4 |- ( ( w e. ( A X. B ) /\ E. x E. y ( w = <. x , y >. /\ ph ) ) <-> E. x E. y ( w = <. x , y >. /\ ( ( x e. A /\ y e. B ) /\ ph ) ) )
12	11	opabbii 4179	. . 3 |- { <. w , z >. | ( w e. ( A X. B ) /\ E. x E. y ( w = <. x , y >. /\ ph ) ) } = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ( ( x e. A /\ y e. B ) /\ ph ) ) }
13	1, 12	eqtri 2255	. 2 |- ( { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) } |` ( A X. B ) ) = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ( ( x e. A /\ y e. B ) /\ ph ) ) }
14		dfoprab2 6102	. . 3 |- { <. <. x , y >. , z >. | ph } = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) }
15	14	reseq1i 5036	. 2 |- ( { <. <. x , y >. , z >. | ph } |` ( A X. B ) ) = ( { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) } |` ( A X. B ) )
16		dfoprab2 6102	. 2 |- { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ph ) } = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ( ( x e. A /\ y e. B ) /\ ph ) ) }
17	13, 15, 16	3eqtr4i 2265	1 |- ( { <. <. x , y >. , z >. | ph } |` ( A X. B ) ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ph ) }

Syntax hints:   /\ wa 104   = wceq 1398   E.wex 1541   e. wcel 2205   <.cop 3694   {copab 4172   X. cxp 4749   |` cres 4753   {coprab 6053

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-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-opab 4174  df-xp 4757  df-rel 4758  df-res 4763  df-oprab 6056

This theorem is referenced by:  resoprab2  6152