Theorem smodm2 7452

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 7448	. 2 |- ( Smo F -> Ord dom F )
2		fndm 5990	. . . 4 |- ( F Fn A -> dom F = A )
3		ordeq 5730	. . . 4 |- ( dom F = A -> ( Ord dom F <-> Ord A ) )
4	2, 3	syl 17	. . 3 |- ( F Fn A -> ( Ord dom F <-> Ord A ) )
5	4	biimpa 501	. 2 |- ( ( F Fn A /\ Ord dom F ) -> Ord A )
6	1, 5	sylan2 491	1 |- ( ( F Fn A /\ Smo F ) -> Ord A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   dom cdm 5114   Ord word 5722   Fn wfn 5883   Smo wsmo 7442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-in 3581  df-ss 3588  df-uni 4437  df-tr 4753  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-fn 5891  df-smo 7443

This theorem is referenced by:  smo11  7461  smoord  7462  smoword  7463  smogt  7464  smorndom  7465  coftr  9095