Theorem resopab2 4938

Description: Restriction of a class abstraction of ordered pairs. (Contributed by NM, 24-Aug-2007.)

Assertion

Ref	Expression
resopab2	|- ( A C_ B -> ( { <. x , y >. | ( x e. B /\ ph ) } |` A ) = { <. x , y >. | ( x e. A /\ ph ) } )

Distinct variable groups:   x, y, A   x, B, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem resopab2

Step	Hyp	Ref	Expression
1		resopab 4935	. 2 |- ( { <. x , y >. | ( x e. B /\ ph ) } |` A ) = { <. x , y >. | ( x e. A /\ ( x e. B /\ ph ) ) }
2		ssel 3141	. . . . . 6 |- ( A C_ B -> ( x e. A -> x e. B ) )
3	2	pm4.71d 391	. . . . 5 |- ( A C_ B -> ( x e. A <-> ( x e. A /\ x e. B ) ) )
4	3	anbi1d 462	. . . 4 |- ( A C_ B -> ( ( x e. A /\ ph ) <-> ( ( x e. A /\ x e. B ) /\ ph ) ) )
5		anass 399	. . . 4 |- ( ( ( x e. A /\ x e. B ) /\ ph ) <-> ( x e. A /\ ( x e. B /\ ph ) ) )
6	4, 5	bitr2di 196	. . 3 |- ( A C_ B -> ( ( x e. A /\ ( x e. B /\ ph ) ) <-> ( x e. A /\ ph ) ) )
7	6	opabbidv 4055	. 2 |- ( A C_ B -> { <. x , y >. | ( x e. A /\ ( x e. B /\ ph ) ) } = { <. x , y >. | ( x e. A /\ ph ) } )
8	1, 7	eqtrid 2215	1 |- ( A C_ B -> ( { <. x , y >. | ( x e. B /\ ph ) } |` A ) = { <. x , y >. | ( x e. A /\ ph ) } )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   C_ wss 3121   {copab 4049   |` cres 4613

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

This theorem is referenced by:  resmpt  4939