Theorem smodm2 6244

Description: The domain of a strictly monotone ordinal function is an ordinal. (Contributed by Mario Carneiro, 12-Mar-2013.)

Assertion

Ref	Expression
smodm2	|- ( ( F Fn A /\ Smo F ) -> Ord A )

Proof of Theorem smodm2

Step	Hyp	Ref	Expression
1		smodm 6240	. 2 |- ( Smo F -> Ord dom F )
2		fndm 5271	. . . 4 |- ( F Fn A -> dom F = A )
3		ordeq 4334	. . . 4 |- ( dom F = A -> ( Ord dom F <-> Ord A ) )
4	2, 3	syl 14	. . 3 |- ( F Fn A -> ( Ord dom F <-> Ord A ) )
5	4	biimpa 294	. 2 |- ( ( F Fn A /\ Ord dom F ) -> Ord A )
6	1, 5	sylan2 284	1 |- ( ( F Fn A /\ Smo F ) -> Ord A )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   Ord word 4324   dom cdm 4588   Fn wfn 5167   Smo wsmo 6234

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-in 3108  df-ss 3115  df-uni 3775  df-tr 4065  df-iord 4328  df-fn 5175  df-smo 6235

This theorem is referenced by: (None)