Theorem smodm2 5910

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 5906	. 2 |- ( Smo F -> Ord dom F )
2		fndm 4998	. . . 4 |- ( F Fn A -> dom F = A )
3		ordeq 4109	. . . 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 280	. 2 |- ( ( F Fn A /\ Ord dom F ) -> Ord A )
6	1, 5	sylan2 270	1 |- ( ( F Fn A /\ Smo F ) -> Ord A )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   Ord word 4099   dom cdm 4345   Fn wfn 4897   Smo wsmo 5900

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-in 2924  df-ss 2931  df-uni 3581  df-tr 3855  df-iord 4103  df-fn 4905  df-smo 5901

This theorem is referenced by: (None)