Theorem smofvon 5914

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 5901	. . 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 919	. 2 |- ( Smo B -> B : dom B --> On )
3	2	ffvelrnda 5302	1 |- ( ( Smo B /\ A e. dom B ) -> ( B ` A ) e. On )

Syntax hints:   -> wi 4   /\ wa 97   e. wcel 1393   A.wral 2306   Ord word 4099   Oncon0 4100   dom cdm 4345   -->wf 4898   ` cfv 4902   Smo wsmo 5900

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fv 4910  df-smo 5901

This theorem is referenced by:  smoiun  5916