Theorem resoprab 6380

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 5318	. . 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 1951	. . . . 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 795	. . . . . . 7 |- ( ( w e. ( A X. B ) /\ ( w = <. x , y >. /\ ph ) ) <-> ( w = <. x , y >. /\ ( w e. ( A X. B ) /\ ph ) ) )
4		eleq1 2539	. . . . . . . . . 10 |- ( w = <. x , y >. -> ( w e. ( A X. B ) <-> <. x , y >. e. ( A X. B ) ) )
5		opelxp 5028	. . . . . . . . . 10 |- ( <. x , y >. e. ( A X. B ) <-> ( x e. A /\ y e. B ) )
6	4, 5	syl6bb 261	. . . . . . . . 9 |- ( w = <. x , y >. -> ( w e. ( A X. B ) <-> ( x e. A /\ y e. B ) ) )
7	6	anbi1d 704	. . . . . . . 8 |- ( w = <. x , y >. -> ( ( w e. ( A X. B ) /\ ph ) <-> ( ( x e. A /\ y e. B ) /\ ph ) ) )
8	7	pm5.32i 637	. . . . . . 7 |- ( ( w = <. x , y >. /\ ( w e. ( A X. B ) /\ ph ) ) <-> ( w = <. x , y >. /\ ( ( x e. A /\ y e. B ) /\ ph ) ) )
9	3, 8	bitri 249	. . . . . 6 |- ( ( w e. ( A X. B ) /\ ( w = <. x , y >. /\ ph ) ) <-> ( w = <. x , y >. /\ ( ( x e. A /\ y e. B ) /\ ph ) ) )
10	9	2exbii 1645	. . . . 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 251	. . . 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 4511	. . 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 2496	. 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 6325	. . 3 |- { <. <. x , y >. , z >. | ph } = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) }
15	14	reseq1i 5267	. 2 |- ( { <. <. x , y >. , z >. | ph } |` ( A X. B ) ) = ( { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) } |` ( A X. B ) )
16		dfoprab2 6325	. 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 2506	1 |- ( { <. <. x , y >. , z >. | ph } |` ( A X. B ) ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ph ) }

Syntax hints:   /\ wa 369   = wceq 1379   E.wex 1596   e. wcel 1767   <.cop 4033   {copab 4504   X. cxp 4997   |` cres 5001   {coprab 6283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-opab 4506  df-xp 5005  df-rel 5006  df-res 5011  df-oprab 6286

This theorem is referenced by:  resoprab2  6381  df1stres  27194  df2ndres  27195