Theorem smofvon 6290

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 6277	. . 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 1012	. 2 |- ( Smo B -> B : dom B --> On )
3	2	ffvelcdmda 5643	1 |- ( ( Smo B /\ A e. dom B ) -> ( B ` A ) e. On )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2146   A.wral 2453   Ord word 4356   Oncon0 4357   dom cdm 4620   -->wf 5204   ` cfv 5208   Smo wsmo 6276

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-sbc 2961  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-fv 5216  df-smo 6277

This theorem is referenced by:  smoiun  6292