Theorem resoprab2 5950

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 5949	. 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 581	. . . . . 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 3141	. . . . . . . . 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 3141	. . . . . . . . 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 601	. . . . . 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 462	. . . 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 5907	. 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 2215	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 1348   e. wcel 2141   C_ wss 3121   X. cxp 4609   |` cres 4613   {coprab 5854

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-opab 4051  df-xp 4617  df-rel 4618  df-res 4623  df-oprab 5857

This theorem is referenced by:  resmpo  5951