Metamath Proof Explorer


Theorem oiiso2

Description: The order isomorphism of the well-order R on A is an isomorphism onto ran O (which is a subset of A by oif ). (Contributed by Mario Carneiro, 25-Jun-2015)

Ref Expression
Hypothesis oicl.1 F = OrdIso R A
Assertion oiiso2 R We A R Se A F Isom E , R dom F ran F

Proof

Step Hyp Ref Expression
1 oicl.1 F = OrdIso R A
2 eqid recs h V ι v w A | j ran h j R w | u w A | j ran h j R w ¬ u R v = recs h V ι v w A | j ran h j R w | u w A | j ran h j R w ¬ u R v
3 eqid w A | j ran h j R w = w A | j ran h j R w
4 eqid h V ι v w A | j ran h j R w | u w A | j ran h j R w ¬ u R v = h V ι v w A | j ran h j R w | u w A | j ran h j R w ¬ u R v
5 2 3 4 ordtypecbv recs f V ι s y A | i ran f i R y | r y A | i ran f i R y ¬ r R s = recs h V ι v w A | j ran h j R w | u w A | j ran h j R w ¬ u R v
6 eqid x On | t A z recs f V ι s y A | i ran f i R y | r y A | i ran f i R y ¬ r R s x z R t = x On | t A z recs f V ι s y A | i ran f i R y | r y A | i ran f i R y ¬ r R s x z R t
7 simpl R We A R Se A R We A
8 simpr R We A R Se A R Se A
9 5 3 4 6 1 7 8 ordtypelem8 R We A R Se A F Isom E , R dom F ran F