Theorem resopab2 4930

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 4927	. 2 |- ( { <. x , y >. | ( x e. B /\ ph ) } |` A ) = { <. x , y >. | ( x e. A /\ ( x e. B /\ ph ) ) }
2		ssel 3135	. . . . . 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	bitr2di 196	. . 3 |- ( A C_ B -> ( ( x e. A /\ ( x e. B /\ ph ) ) <-> ( x e. A /\ ph ) ) )
7	6	opabbidv 4047	. 2 |- ( A C_ B -> { <. x , y >. | ( x e. A /\ ( x e. B /\ ph ) ) } = { <. x , y >. | ( x e. A /\ ph ) } )
8	1, 7	syl5eq 2210	1 |- ( A C_ B -> ( { <. x , y >. | ( x e. B /\ ph ) } |` A ) = { <. x , y >. | ( x e. A /\ ph ) } )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   C_ wss 3115   {copab 4041   |` cres 4605

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4099  ax-pow 4152  ax-pr 4186

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ral 2448  df-rex 2449  df-v 2727  df-un 3119  df-in 3121  df-ss 3128  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-opab 4043  df-xp 4609  df-rel 4610  df-res 4615

This theorem is referenced by:  resmpt  4931