Theorem resoprab2 6023

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 6022	. 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 3178	. . . . . . . . 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 3178	. . . . . . . . 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 5980	. 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 2241	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 1364   e. wcel 2167   C_ wss 3157   X. cxp 4662   |` cres 4666   {coprab 5926

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-opab 4096  df-xp 4670  df-rel 4671  df-res 4676  df-oprab 5929

This theorem is referenced by:  resmpo  6024