Theorem smofvon 6445

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 6432	. . 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 1036	. 2 |- ( Smo B -> B : dom B --> On )
3	2	ffvelcdmda 5770	1 |- ( ( Smo B /\ A e. dom B ) -> ( B ` A ) e. On )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2200   A.wral 2508   Ord word 4453   Oncon0 4454   dom cdm 4719   -->wf 5314   ` cfv 5318   Smo wsmo 6431

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-smo 6432

This theorem is referenced by:  smoiun  6447