Theorem smofvon 6267

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

Assertion

Ref	Expression
smofvon	|- ( ( Smo B /\ A e. dom B ) -> ( B ` A ) e. On )

Proof of Theorem smofvon

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

Step	Hyp	Ref	Expression
1		df-smo 6254	. . 3 |- ( Smo B <-> ( B : dom B --> On /\ Ord dom B /\ A. x e. dom B A. y e. dom B ( x e. y -> ( B ` x ) e. ( B ` y ) ) ) )
2	1	simp1bi 1002	. 2 |- ( Smo B -> B : dom B --> On )
3	2	ffvelrnda 5620	1 |- ( ( Smo B /\ A e. dom B ) -> ( B ` A ) e. On )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2136   A.wral 2444   Ord word 4340   Oncon0 4341   dom cdm 4604   -->wf 5184   ` cfv 5188   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-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-smo 6254

This theorem is referenced by:  smoiun  6269