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 ) )