Theorem resoprab 6379

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 5128	. . 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 1839	. . . . 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 811	. . . . . . 7 |- ( ( w e. ( A X. B ) /\ ( w = <. x , y >. /\ ph ) ) <-> ( w = <. x , y >. /\ ( w e. ( A X. B ) /\ ph ) ) )
4		eleq1 2517	. . . . . . . . . 10 |- ( w = <. x , y >. -> ( w e. ( A X. B ) <-> <. x , y >. e. ( A X. B ) ) )
5		opelxp 4841	. . . . . . . . . 10 |- ( <. x , y >. e. ( A X. B ) <-> ( x e. A /\ y e. B ) )
6	4, 5	syl6bb 269	. . . . . . . . 9 |- ( w = <. x , y >. -> ( w e. ( A X. B ) <-> ( x e. A /\ y e. B ) ) )
7	6	anbi1d 716	. . . . . . . 8 |- ( w = <. x , y >. -> ( ( w e. ( A X. B ) /\ ph ) <-> ( ( x e. A /\ y e. B ) /\ ph ) ) )
8	7	pm5.32i 647	. . . . . . 7 |- ( ( w = <. x , y >. /\ ( w e. ( A X. B ) /\ ph ) ) <-> ( w = <. x , y >. /\ ( ( x e. A /\ y e. B ) /\ ph ) ) )
9	3, 8	bitri 257	. . . . . 6 |- ( ( w e. ( A X. B ) /\ ( w = <. x , y >. /\ ph ) ) <-> ( w = <. x , y >. /\ ( ( x e. A /\ y e. B ) /\ ph ) ) )
10	9	2exbii 1722	. . . . 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 259	. . . 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 4438	. . 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 2473	. 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 6324	. . 3 |- { <. <. x , y >. , z >. | ph } = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) }
15	14	reseq1i 5078	. 2 |- ( { <. <. x , y >. , z >. | ph } |` ( A X. B ) ) = ( { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) } |` ( A X. B ) )
16		dfoprab2 6324	. 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 2483	1 |- ( { <. <. x , y >. , z >. | ph } |` ( A X. B ) ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ph ) }

Syntax hints:   /\ wa 375   = wceq 1447   E.wex 1666   e. wcel 1890   <.cop 3941   {copab 4431   X. cxp 4809   |` cres 4813   {coprab 6276

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-9 1899  ax-10 1918  ax-11 1923  ax-12 1936  ax-13 2091  ax-ext 2431  ax-sep 4496  ax-nul 4505  ax-pr 4611

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1450  df-ex 1667  df-nf 1671  df-sb 1801  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3014  df-dif 3374  df-un 3376  df-in 3378  df-ss 3385  df-nul 3699  df-if 3849  df-sn 3936  df-pr 3938  df-op 3942  df-opab 4433  df-xp 4817  df-rel 4818  df-res 4823  df-oprab 6279

This theorem is referenced by:  resoprab2  6380  df1stres  28291  df2ndres  28292