Metamath Proof Explorer

Theorem bnj1373

Description: Technical lemma for bnj60 . This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Hypotheses bnj1373.1 
|- B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
bnj1373.2 
|- Y = <. x , ( f |` _pred ( x , A , R ) ) >.
bnj1373.3 
|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1373.4 
|- ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
bnj1373.5 
|- ( ta' <-> [. y / x ]. ta )
Assertion bnj1373 
|- ( ta' <-> ( f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) )

Proof

Step Hyp Ref Expression
1 bnj1373.1 
 |-  B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
2 bnj1373.2 
 |-  Y = <. x , ( f |` _pred ( x , A , R ) ) >.
3 bnj1373.3 
 |-  C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
4 bnj1373.4 
 |-  ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
5 bnj1373.5 
 |-  ( ta' <-> [. y / x ]. ta )
6 1 bnj1309 
 |-  ( f e. B -> A. x f e. B )
7 3 6 bnj1307 
 |-  ( f e. C -> A. x f e. C )
8 7 bnj1351 
 |-  ( ( f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) -> A. x ( f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) )
9 8 nf5i 
 |-  F/ x ( f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) )
10 sneq 
 |-  ( x = y -> { x } = { y } )
11 bnj1318 
 |-  ( x = y -> _trCl ( x , A , R ) = _trCl ( y , A , R ) )
12 10 11 uneq12d 
 |-  ( x = y -> ( { x } u. _trCl ( x , A , R ) ) = ( { y } u. _trCl ( y , A , R ) ) )
13 12 eqeq2d 
 |-  ( x = y -> ( dom f = ( { x } u. _trCl ( x , A , R ) ) <-> dom f = ( { y } u. _trCl ( y , A , R ) ) ) )
14 13 anbi2d 
 |-  ( x = y -> ( ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) <-> ( f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) ) )
15 4 14 syl5bb 
 |-  ( x = y -> ( ta <-> ( f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) ) )
16 9 15 sbciegf 
 |-  ( y e. _V -> ( [. y / x ]. ta <-> ( f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) ) )
17 16 elv 
 |-  ( [. y / x ]. ta <-> ( f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) )
18 5 17 bitri 
 |-  ( ta' <-> ( f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) )