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Metamath Proof Explorer

Theorem ordtype2

Description: For any set-like well-ordered class, if the order isomorphism exists (is a set), then it maps some ordinal onto A isomorphically. Otherwise, F is a proper class, which implies that either ran F C_ A is a proper class or dom F = On . This weak version of ordtype does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 25-Jun-2015)

Ref Expression
Hypothesis oicl.1 
|- F = OrdIso ( R , A )
Assertion ordtype2 
|- ( ( R We A /\ R Se A /\ F e. _V ) -> F Isom _E , R ( dom F , A ) )

Proof

Step Hyp Ref Expression
1 oicl.1 
 |-  F = OrdIso ( R , A )
2 eqid 
 |-  recs ( ( h e. _V |-> ( iota_ v e. { w e. A | A. j e. ran h j R w } A. u e. { w e. A | A. j e. ran h j R w } -. u R v ) ) ) = recs ( ( h e. _V |-> ( iota_ v e. { w e. A | A. j e. ran h j R w } A. u e. { w e. A | A. j e. ran h j R w } -. u R v ) ) )
3 eqid 
 |-  { w e. A | A. j e. ran h j R w } = { w e. A | A. j e. ran h j R w }
4 eqid 
 |-  ( h e. _V |-> ( iota_ v e. { w e. A | A. j e. ran h j R w } A. u e. { w e. A | A. j e. ran h j R w } -. u R v ) ) = ( h e. _V |-> ( iota_ v e. { w e. A | A. j e. ran h j R w } A. u e. { w e. A | A. j e. ran h j R w } -. u R v ) )
5 2 3 4 ordtypecbv 
 |-  recs ( ( f e. _V |-> ( iota_ s e. { y e. A | A. i e. ran f i R y } A. r e. { y e. A | A. i e. ran f i R y } -. r R s ) ) ) = recs ( ( h e. _V |-> ( iota_ v e. { w e. A | A. j e. ran h j R w } A. u e. { w e. A | A. j e. ran h j R w } -. u R v ) ) )
6 eqid 
 |-  { x e. On | E. t e. A A. z e. ( recs ( ( f e. _V |-> ( iota_ s e. { y e. A | A. i e. ran f i R y } A. r e. { y e. A | A. i e. ran f i R y } -. r R s ) ) ) " x ) z R t } = { x e. On | E. t e. A A. z e. ( recs ( ( f e. _V |-> ( iota_ s e. { y e. A | A. i e. ran f i R y } A. r e. { y e. A | A. i e. ran f i R y } -. r R s ) ) ) " x ) z R t }
7 simp1 
 |-  ( ( R We A /\ R Se A /\ F e. _V ) -> R We A )
8 simp2 
 |-  ( ( R We A /\ R Se A /\ F e. _V ) -> R Se A )
9 simp3 
 |-  ( ( R We A /\ R Se A /\ F e. _V ) -> F e. _V )
10 5 3 4 6 1 7 8 9 ordtypelem9 
 |-  ( ( R We A /\ R Se A /\ F e. _V ) -> F Isom _E , R ( dom F , A ) )