Theorem smofvon 7049

Description: If is a strictly monotone ordinal function, and is in the domain of , then the value of the function at is an ordinal. (Contributed by Andrew Salmon, 20-Nov-2011.)

Assertion

Ref	Expression
smofvon

Proof of Theorem smofvon

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-smo 7036	. . 3
2	1	simp1bi 1011	. 2
3	2	ffvelrnda 6031	1

Syntax hints:  ->wi 4  /\wa 369  e.wcel 1818  A.wral 2807  Ordword 4882  con0 4883  domcdm 5004  -->wf 5589  `cfv 5593  Smowsmo 7035

This theorem is referenced by:  smoiun  7051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-fv 5601  df-smo 7036