Theorem resoprab 6371

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 5308	. . 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 1782	. . . . 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 2526	. . . . . . . . . 10 |- ( w = <. x , y >. -> ( w e. ( A X. B ) <-> <. x , y >. e. ( A X. B ) ) )
5		opelxp 5018	. . . . . . . . . 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 702	. . . . . . . 8 |- ( w = <. x , y >. -> ( ( w e. ( A X. B ) /\ ph ) <-> ( ( x e. A /\ y e. B ) /\ ph ) ) )
8	7	pm5.32i 635	. . . . . . 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 1673	. . . . 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 4503	. . 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 2483	. 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 6316	. . 3 |- { <. <. x , y >. , z >. | ph } = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) }
15	14	reseq1i 5258	. 2 |- ( { <. <. x , y >. , z >. | ph } |` ( A X. B ) ) = ( { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) } |` ( A X. B ) )
16		dfoprab2 6316	. 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 2493	1 |- ( { <. <. x , y >. , z >. | ph } |` ( A X. B ) ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ph ) }

Syntax hints:   /\ wa 367   = wceq 1398   E.wex 1617   e. wcel 1823   <.cop 4022   {copab 4496   X. cxp 4986   |` cres 4990   {coprab 6271

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-opab 4498  df-xp 4994  df-rel 4995  df-res 5000  df-oprab 6274

This theorem is referenced by:  resoprab2  6372  df1stres  27750  df2ndres  27751