Theorem resopab2 5090

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 5087	. 2 |- ( { <. x , y >. | ( x e. B /\ ph ) } |` A ) = { <. x , y >. | ( x e. A /\ ( x e. B /\ ph ) ) }
2		ssel 3236	. . . . . 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 4181	. 2 |- ( A C_ B -> { <. x , y >. | ( x e. A /\ ( x e. B /\ ph ) ) } = { <. x , y >. | ( x e. A /\ ph ) } )
8	1, 7	eqtrid 2279	1 |- ( A C_ B -> ( { <. x , y >. | ( x e. B /\ ph ) } |` A ) = { <. x , y >. | ( x e. A /\ ph ) } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   C_ wss 3214   {copab 4175   |` cres 4756

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-opab 4177  df-xp 4760  df-rel 4761  df-res 4766

This theorem is referenced by:  resmpt  5091