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Metamath Proof Explorer

Theorem resoprab

Description: Restriction of an operation class abstraction. (Contributed by NM, 10-Feb-2007)

Ref Expression
Assertion resoprab 
|- ( { <. <. x , y >. , z >. | ph } |` ( A X. B ) ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ph ) }

Proof

Step Hyp Ref Expression
1 resopab 
 |-  ( { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) } |` ( A X. B ) ) = { <. w , z >. | ( w e. ( A X. B ) /\ E. x E. y ( w = <. x , y >. /\ ph ) ) }
2 19.42vv 
 |-  ( E. x E. y ( w e. ( A X. B ) /\ ( w = <. x , y >. /\ ph ) ) <-> ( w e. ( A X. B ) /\ E. x E. y ( w = <. x , y >. /\ ph ) ) )
3 an12 
 |-  ( ( w e. ( A X. B ) /\ ( w = <. x , y >. /\ ph ) ) <-> ( w = <. x , y >. /\ ( w e. ( A X. B ) /\ ph ) ) )
4 eleq1 
 |-  ( w = <. x , y >. -> ( w e. ( A X. B ) <-> <. x , y >. e. ( A X. B ) ) )
5 opelxp 
 |-  ( <. x , y >. e. ( A X. B ) <-> ( x e. A /\ y e. B ) )
6 4 5 bitrdi 
 |-  ( w = <. x , y >. -> ( w e. ( A X. B ) <-> ( x e. A /\ y e. B ) ) )
7 6 anbi1d 
 |-  ( w = <. x , y >. -> ( ( w e. ( A X. B ) /\ ph ) <-> ( ( x e. A /\ y e. B ) /\ ph ) ) )
8 7 pm5.32i 
 |-  ( ( w = <. x , y >. /\ ( w e. ( A X. B ) /\ ph ) ) <-> ( w = <. x , y >. /\ ( ( x e. A /\ y e. B ) /\ ph ) ) )
9 3 8 bitri 
 |-  ( ( w e. ( A X. B ) /\ ( w = <. x , y >. /\ ph ) ) <-> ( w = <. x , y >. /\ ( ( x e. A /\ y e. B ) /\ ph ) ) )
10 9 2exbii 
 |-  ( E. x E. y ( w e. ( A X. B ) /\ ( w = <. x , y >. /\ ph ) ) <-> E. x E. y ( w = <. x , y >. /\ ( ( x e. A /\ y e. B ) /\ ph ) ) )
11 2 10 bitr3i 
 |-  ( ( w e. ( A X. B ) /\ E. x E. y ( w = <. x , y >. /\ ph ) ) <-> E. x E. y ( w = <. x , y >. /\ ( ( x e. A /\ y e. B ) /\ ph ) ) )
12 11 opabbii 
 |-  { <. w , z >. | ( w e. ( A X. B ) /\ E. x E. y ( w = <. x , y >. /\ ph ) ) } = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ( ( x e. A /\ y e. B ) /\ ph ) ) }
13 1 12 eqtri 
 |-  ( { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) } |` ( A X. B ) ) = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ( ( x e. A /\ y e. B ) /\ ph ) ) }
14 dfoprab2 
 |-  { <. <. x , y >. , z >. | ph } = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) }
15 14 reseq1i 
 |-  ( { <. <. x , y >. , z >. | ph } |` ( A X. B ) ) = ( { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) } |` ( A X. B ) )
16 dfoprab2 
 |-  { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ph ) } = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ( ( x e. A /\ y e. B ) /\ ph ) ) }
17 13 15 16 3eqtr4i 
 |-  ( { <. <. x , y >. , z >. | ph } |` ( A X. B ) ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ph ) }