Theorem resopab2 5003

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 5000	. 2 |- ( { <. x , y >. | ( x e. B /\ ph ) } |` A ) = { <. x , y >. | ( x e. A /\ ( x e. B /\ ph ) ) }
2		ssel 3186	. . . . . 6 |- ( A C_ B -> ( x e. A -> x e. B ) )
3	2	pm4.71d 393	. . . . 5 |- ( A C_ B -> ( x e. A <-> ( x e. A /\ x e. B ) ) )
4	3	anbi1d 465	. . . 4 |- ( A C_ B -> ( ( x e. A /\ ph ) <-> ( ( x e. A /\ x e. B ) /\ ph ) ) )
5		anass 401	. . . 4 |- ( ( ( x e. A /\ x e. B ) /\ ph ) <-> ( x e. A /\ ( x e. B /\ ph ) ) )
6	4, 5	bitr2di 197	. . 3 |- ( A C_ B -> ( ( x e. A /\ ( x e. B /\ ph ) ) <-> ( x e. A /\ ph ) ) )
7	6	opabbidv 4109	. 2 |- ( A C_ B -> { <. x , y >. | ( x e. A /\ ( x e. B /\ ph ) ) } = { <. x , y >. | ( x e. A /\ ph ) } )
8	1, 7	eqtrid 2249	1 |- ( A C_ B -> ( { <. x , y >. | ( x e. B /\ ph ) } |` A ) = { <. x , y >. | ( x e. A /\ ph ) } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1372   e. wcel 2175   C_ wss 3165   {copab 4103   |` cres 4675

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-opab 4105  df-xp 4679  df-rel 4680  df-res 4685

This theorem is referenced by:  resmpt  5004