Theorem smodm2 6504

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 6500	. 2 |- ( Smo F -> Ord dom F )
2		fndm 5436	. . . 4 |- ( F Fn A -> dom F = A )
3		ordeq 4475	. . . 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 296	. 2 |- ( ( F Fn A /\ Ord dom F ) -> Ord A )
6	1, 5	sylan2 286	1 |- ( ( F Fn A /\ Smo F ) -> Ord A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   Ord word 4465   dom cdm 4731   Fn wfn 5328   Smo wsmo 6494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-in 3207  df-ss 3214  df-uni 3899  df-tr 4193  df-iord 4469  df-fn 5336  df-smo 6495

This theorem is referenced by: (None)