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Metamath Proof Explorer

Theorem ordtypelem1

Description: Lemma for ordtype . (Contributed by Mario Carneiro, 24-Jun-2015)

Ref Expression
Hypotheses ordtypelem.1 
|- F = recs ( G )
ordtypelem.2 
|- C = { w e. A | A. j e. ran h j R w }
ordtypelem.3 
|- G = ( h e. _V |-> ( iota_ v e. C A. u e. C -. u R v ) )
ordtypelem.5 
|- T = { x e. On | E. t e. A A. z e. ( F " x ) z R t }
ordtypelem.6 
|- O = OrdIso ( R , A )
ordtypelem.7 
|- ( ph -> R We A )
ordtypelem.8 
|- ( ph -> R Se A )
Assertion ordtypelem1 
|- ( ph -> O = ( F |` T ) )

Proof

Step Hyp Ref Expression
1 ordtypelem.1 
 |-  F = recs ( G )
2 ordtypelem.2 
 |-  C = { w e. A | A. j e. ran h j R w }
3 ordtypelem.3 
 |-  G = ( h e. _V |-> ( iota_ v e. C A. u e. C -. u R v ) )
4 ordtypelem.5 
 |-  T = { x e. On | E. t e. A A. z e. ( F " x ) z R t }
5 ordtypelem.6 
 |-  O = OrdIso ( R , A )
6 ordtypelem.7 
 |-  ( ph -> R We A )
7 ordtypelem.8 
 |-  ( ph -> R Se A )
8 iftrue 
 |-  ( ( R We A /\ R Se A ) -> if ( ( R We A /\ R Se A ) , ( F |` { x e. On | E. t e. A A. z e. ( F " x ) z R t } ) , (/) ) = ( F |` { x e. On | E. t e. A A. z e. ( F " x ) z R t } ) )
9 6 7 8 syl2anc 
 |-  ( ph -> if ( ( R We A /\ R Se A ) , ( F |` { x e. On | E. t e. A A. z e. ( F " x ) z R t } ) , (/) ) = ( F |` { x e. On | E. t e. A A. z e. ( F " x ) z R t } ) )
10 2 3 1 dfoi 
 |-  OrdIso ( R , A ) = if ( ( R We A /\ R Se A ) , ( F |` { x e. On | E. t e. A A. z e. ( F " x ) z R t } ) , (/) )
11 5 10 eqtri 
 |-  O = if ( ( R We A /\ R Se A ) , ( F |` { x e. On | E. t e. A A. z e. ( F " x ) z R t } ) , (/) )
12 4 reseq2i 
 |-  ( F |` T ) = ( F |` { x e. On | E. t e. A A. z e. ( F " x ) z R t } )
13 9 11 12 3eqtr4g 
 |-  ( ph -> O = ( F |` T ) )