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Metamath Proof Explorer

Theorem dfoprab2

Description: Class abstraction for operations in terms of class abstraction of ordered pairs. (Contributed by NM, 12-Mar-1995)

Ref Expression
Assertion dfoprab2 
|- { <. <. x , y >. , z >. | ph } = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) }

Proof

Step Hyp Ref Expression
1 excom 
 |-  ( E. z E. w E. x E. y ( v = <. w , z >. /\ ( w = <. x , y >. /\ ph ) ) <-> E. w E. z E. x E. y ( v = <. w , z >. /\ ( w = <. x , y >. /\ ph ) ) )
2 exrot4 
 |-  ( E. z E. w E. x E. y ( v = <. w , z >. /\ ( w = <. x , y >. /\ ph ) ) <-> E. x E. y E. z E. w ( v = <. w , z >. /\ ( w = <. x , y >. /\ ph ) ) )
3 opeq1 
 |-  ( w = <. x , y >. -> <. w , z >. = <. <. x , y >. , z >. )
4 3 eqeq2d 
 |-  ( w = <. x , y >. -> ( v = <. w , z >. <-> v = <. <. x , y >. , z >. ) )
5 4 pm5.32ri 
 |-  ( ( v = <. w , z >. /\ w = <. x , y >. ) <-> ( v = <. <. x , y >. , z >. /\ w = <. x , y >. ) )
6 5 anbi1i 
 |-  ( ( ( v = <. w , z >. /\ w = <. x , y >. ) /\ ph ) <-> ( ( v = <. <. x , y >. , z >. /\ w = <. x , y >. ) /\ ph ) )
7 anass 
 |-  ( ( ( v = <. w , z >. /\ w = <. x , y >. ) /\ ph ) <-> ( v = <. w , z >. /\ ( w = <. x , y >. /\ ph ) ) )
8 an32 
 |-  ( ( ( v = <. <. x , y >. , z >. /\ w = <. x , y >. ) /\ ph ) <-> ( ( v = <. <. x , y >. , z >. /\ ph ) /\ w = <. x , y >. ) )
9 6 7 8 3bitr3i 
 |-  ( ( v = <. w , z >. /\ ( w = <. x , y >. /\ ph ) ) <-> ( ( v = <. <. x , y >. , z >. /\ ph ) /\ w = <. x , y >. ) )
10 9 exbii 
 |-  ( E. w ( v = <. w , z >. /\ ( w = <. x , y >. /\ ph ) ) <-> E. w ( ( v = <. <. x , y >. , z >. /\ ph ) /\ w = <. x , y >. ) )
11 opex 
 |-  <. x , y >. e. _V
12 11 isseti 
 |-  E. w w = <. x , y >.
13 19.42v 
 |-  ( E. w ( ( v = <. <. x , y >. , z >. /\ ph ) /\ w = <. x , y >. ) <-> ( ( v = <. <. x , y >. , z >. /\ ph ) /\ E. w w = <. x , y >. ) )
14 12 13 mpbiran2 
 |-  ( E. w ( ( v = <. <. x , y >. , z >. /\ ph ) /\ w = <. x , y >. ) <-> ( v = <. <. x , y >. , z >. /\ ph ) )
15 10 14 bitri 
 |-  ( E. w ( v = <. w , z >. /\ ( w = <. x , y >. /\ ph ) ) <-> ( v = <. <. x , y >. , z >. /\ ph ) )
16 15 3exbii 
 |-  ( E. x E. y E. z E. w ( v = <. w , z >. /\ ( w = <. x , y >. /\ ph ) ) <-> E. x E. y E. z ( v = <. <. x , y >. , z >. /\ ph ) )
17 2 16 bitri 
 |-  ( E. z E. w E. x E. y ( v = <. w , z >. /\ ( w = <. x , y >. /\ ph ) ) <-> E. x E. y E. z ( v = <. <. x , y >. , z >. /\ ph ) )
18 19.42vv 
 |-  ( E. x E. y ( v = <. w , z >. /\ ( w = <. x , y >. /\ ph ) ) <-> ( v = <. w , z >. /\ E. x E. y ( w = <. x , y >. /\ ph ) ) )
19 18 2exbii 
 |-  ( E. w E. z E. x E. y ( v = <. w , z >. /\ ( w = <. x , y >. /\ ph ) ) <-> E. w E. z ( v = <. w , z >. /\ E. x E. y ( w = <. x , y >. /\ ph ) ) )
20 1 17 19 3bitr3i 
 |-  ( E. x E. y E. z ( v = <. <. x , y >. , z >. /\ ph ) <-> E. w E. z ( v = <. w , z >. /\ E. x E. y ( w = <. x , y >. /\ ph ) ) )
21 20 abbii 
 |-  { v | E. x E. y E. z ( v = <. <. x , y >. , z >. /\ ph ) } = { v | E. w E. z ( v = <. w , z >. /\ E. x E. y ( w = <. x , y >. /\ ph ) ) }
22 df-oprab 
 |-  { <. <. x , y >. , z >. | ph } = { v | E. x E. y E. z ( v = <. <. x , y >. , z >. /\ ph ) }
23 df-opab 
 |-  { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) } = { v | E. w E. z ( v = <. w , z >. /\ E. x E. y ( w = <. x , y >. /\ ph ) ) }
24 21 22 23 3eqtr4i 
 |-  { <. <. x , y >. , z >. | ph } = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) }