Theorem resoprab2 5974

Description: Restriction of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
resoprab2	|- ( ( C C_ A /\ D C_ B ) -> ( { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ph ) } |` ( C X. D ) ) = { <. <. x , y >. , z >. | ( ( x e. C /\ y e. D ) /\ ph ) } )

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

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

Proof of Theorem resoprab2

Step	Hyp	Ref	Expression
1		resoprab 5973	. 2 |- ( { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ph ) } |` ( C X. D ) ) = { <. <. x , y >. , z >. | ( ( x e. C /\ y e. D ) /\ ( ( x e. A /\ y e. B ) /\ ph ) ) }
2		anass 401	. . . 4 |- ( ( ( ( x e. C /\ y e. D ) /\ ( x e. A /\ y e. B ) ) /\ ph ) <-> ( ( x e. C /\ y e. D ) /\ ( ( x e. A /\ y e. B ) /\ ph ) ) )
3		an4 586	. . . . . 6 |- ( ( ( x e. C /\ y e. D ) /\ ( x e. A /\ y e. B ) ) <-> ( ( x e. C /\ x e. A ) /\ ( y e. D /\ y e. B ) ) )
4		ssel 3151	. . . . . . . . 9 |- ( C C_ A -> ( x e. C -> x e. A ) )
5	4	pm4.71d 393	. . . . . . . 8 |- ( C C_ A -> ( x e. C <-> ( x e. C /\ x e. A ) ) )
6	5	bicomd 141	. . . . . . 7 |- ( C C_ A -> ( ( x e. C /\ x e. A ) <-> x e. C ) )
7		ssel 3151	. . . . . . . . 9 |- ( D C_ B -> ( y e. D -> y e. B ) )
8	7	pm4.71d 393	. . . . . . . 8 |- ( D C_ B -> ( y e. D <-> ( y e. D /\ y e. B ) ) )
9	8	bicomd 141	. . . . . . 7 |- ( D C_ B -> ( ( y e. D /\ y e. B ) <-> y e. D ) )
10	6, 9	bi2anan9 606	. . . . . 6 |- ( ( C C_ A /\ D C_ B ) -> ( ( ( x e. C /\ x e. A ) /\ ( y e. D /\ y e. B ) ) <-> ( x e. C /\ y e. D ) ) )
11	3, 10	bitrid 192	. . . . 5 |- ( ( C C_ A /\ D C_ B ) -> ( ( ( x e. C /\ y e. D ) /\ ( x e. A /\ y e. B ) ) <-> ( x e. C /\ y e. D ) ) )
12	11	anbi1d 465	. . . 4 |- ( ( C C_ A /\ D C_ B ) -> ( ( ( ( x e. C /\ y e. D ) /\ ( x e. A /\ y e. B ) ) /\ ph ) <-> ( ( x e. C /\ y e. D ) /\ ph ) ) )
13	2, 12	bitr3id 194	. . 3 |- ( ( C C_ A /\ D C_ B ) -> ( ( ( x e. C /\ y e. D ) /\ ( ( x e. A /\ y e. B ) /\ ph ) ) <-> ( ( x e. C /\ y e. D ) /\ ph ) ) )
14	13	oprabbidv 5931	. 2 |- ( ( C C_ A /\ D C_ B ) -> { <. <. x , y >. , z >. | ( ( x e. C /\ y e. D ) /\ ( ( x e. A /\ y e. B ) /\ ph ) ) } = { <. <. x , y >. , z >. | ( ( x e. C /\ y e. D ) /\ ph ) } )
15	1, 14	eqtrid 2222	1 |- ( ( C C_ A /\ D C_ B ) -> ( { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ph ) } |` ( C X. D ) ) = { <. <. x , y >. , z >. | ( ( x e. C /\ y e. D ) /\ ph ) } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   C_ wss 3131   X. cxp 4626   |` cres 4630   {coprab 5878

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-opab 4067  df-xp 4634  df-rel 4635  df-res 4640  df-oprab 5881

This theorem is referenced by:  resmpo  5975