Theorem smodm2 6460

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 6456	. 2 |- ( Smo F -> Ord dom F )
2		fndm 5429	. . . 4 |- ( F Fn A -> dom F = A )
3		ordeq 4469	. . . 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 1397   Ord word 4459   dom cdm 4725   Fn wfn 5321   Smo wsmo 6450

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-in 3206  df-ss 3213  df-uni 3894  df-tr 4188  df-iord 4463  df-fn 5329  df-smo 6451

This theorem is referenced by: (None)