Theorem resopab2 4874

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 4871	. 2 |- ( { <. x , y >. | ( x e. B /\ ph ) } |` A ) = { <. x , y >. | ( x e. A /\ ( x e. B /\ ph ) ) }
2		ssel 3096	. . . . . 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 461	. . . 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	syl6rbb 196	. . 3 |- ( A C_ B -> ( ( x e. A /\ ( x e. B /\ ph ) ) <-> ( x e. A /\ ph ) ) )
7	6	opabbidv 4002	. 2 |- ( A C_ B -> { <. x , y >. | ( x e. A /\ ( x e. B /\ ph ) ) } = { <. x , y >. | ( x e. A /\ ph ) } )
8	1, 7	syl5eq 2185	1 |- ( A C_ B -> ( { <. x , y >. | ( x e. B /\ ph ) } |` A ) = { <. x , y >. | ( x e. A /\ ph ) } )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   e. wcel 1481   C_ wss 3076   {copab 3996   |` cres 4549

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

This theorem is referenced by:  resmpt  4875