Theorem resoprab2 5876

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 5875	. 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 399	. . . 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 576	. . . . . 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 3096	. . . . . . . . 9 |- ( C C_ A -> ( x e. C -> x e. A ) )
5	4	pm4.71d 391	. . . . . . . 8 |- ( C C_ A -> ( x e. C <-> ( x e. C /\ x e. A ) ) )
6	5	bicomd 140	. . . . . . 7 |- ( C C_ A -> ( ( x e. C /\ x e. A ) <-> x e. C ) )
7		ssel 3096	. . . . . . . . 9 |- ( D C_ B -> ( y e. D -> y e. B ) )
8	7	pm4.71d 391	. . . . . . . 8 |- ( D C_ B -> ( y e. D <-> ( y e. D /\ y e. B ) ) )
9	8	bicomd 140	. . . . . . 7 |- ( D C_ B -> ( ( y e. D /\ y e. B ) <-> y e. D ) )
10	6, 9	bi2anan9 596	. . . . . 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	syl5bb 191	. . . . 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 461	. . . 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 193	. . 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 5833	. 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	syl5eq 2185	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 103   = wceq 1332   e. wcel 1481   C_ wss 3076   X. cxp 4545   |` cres 4549   {coprab 5783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-opab 3998  df-xp 4553  df-rel 4554  df-res 4559  df-oprab 5786

This theorem is referenced by:  resmpo  5877