Theorem smofvon2dm 6505

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 6496	. . 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 1039	. 2 |- ( Smo F -> F : dom F --> On )
3	2	ffvelcdmda 5790	1 |- ( ( Smo F /\ B e. dom F ) -> ( F ` B ) e. On )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2202   A.wral 2511   Ord word 4465   Oncon0 4466   dom cdm 4731   -->wf 5329   ` cfv 5333   Smo wsmo 6494

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-tr 4193  df-id 4396  df-iord 4469  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-smo 6495

This theorem is referenced by: (None)