Theorem smofvon2dm 6297

Description: The function values of a strictly monotone ordinal function are ordinals. (Contributed by Mario Carneiro, 12-Mar-2013.)

Assertion

Ref	Expression
smofvon2dm	|- ( ( Smo F /\ B e. dom F ) -> ( F ` B ) e. On )

Proof of Theorem smofvon2dm

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

Step	Hyp	Ref	Expression
1		dfsmo2 6288	. . 3 |- ( Smo F <-> ( F : dom F --> On /\ Ord dom F /\ A. x e. dom F A. y e. x ( F ` y ) e. ( F ` x ) ) )
2	1	simp1bi 1012	. 2 |- ( Smo F -> F : dom F --> On )
3	2	ffvelcdmda 5652	1 |- ( ( Smo F /\ B e. dom F ) -> ( F ` B ) e. On )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2148   A.wral 2455   Ord word 4363   Oncon0 4364   dom cdm 4627   -->wf 5213   ` cfv 5217   Smo wsmo 6286

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-sbc 2964  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-tr 4103  df-id 4294  df-iord 4367  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fv 5225  df-smo 6287

This theorem is referenced by: (None)