Theorem smofvon 6236

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 6223	. . 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 997	. 2 |- ( Smo B -> B : dom B --> On )
3	2	ffvelrnda 5595	1 |- ( ( Smo B /\ A e. dom B ) -> ( B ` A ) e. On )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2125   A.wral 2432   Ord word 4317   Oncon0 4318   dom cdm 4579   -->wf 5159   ` cfv 5163   Smo wsmo 6222

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-v 2711  df-sbc 2934  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-br 3962  df-opab 4022  df-id 4248  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-iota 5128  df-fun 5165  df-fn 5166  df-f 5167  df-fv 5171  df-smo 6223

This theorem is referenced by:  smoiun  6238