Theorem smodm2 6263

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 6259	. 2 |- ( Smo F -> Ord dom F )
2		fndm 5287	. . . 4 |- ( F Fn A -> dom F = A )
3		ordeq 4350	. . . 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 1343   Ord word 4340   dom cdm 4604   Fn wfn 5183   Smo wsmo 6253

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-in 3122  df-ss 3129  df-uni 3790  df-tr 4081  df-iord 4344  df-fn 5191  df-smo 6254

This theorem is referenced by: (None)