Theorem smofvon2 7453

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

Assertion

Ref	Expression
smofvon2	|- ( Smo F -> ( F ` B ) e. On )

Proof of Theorem smofvon2

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

Step	Hyp	Ref	Expression
1		dfsmo2 7444	. . . 4 |- ( 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 1076	. . 3 |- ( Smo F -> F : dom F --> On )
3		ffvelrn 6357	. . . 4 |- ( ( F : dom F --> On /\ B e. dom F ) -> ( F ` B ) e. On )
4	3	expcom 451	. . 3 |- ( B e. dom F -> ( F : dom F --> On -> ( F ` B ) e. On ) )
5	2, 4	syl5 34	. 2 |- ( B e. dom F -> ( Smo F -> ( F ` B ) e. On ) )
6		ndmfv 6218	. . . 4 |- ( -. B e. dom F -> ( F ` B ) = (/) )
7		0elon 5778	. . . 4 |- (/) e. On
8	6, 7	syl6eqel 2709	. . 3 |- ( -. B e. dom F -> ( F ` B ) e. On )
9	8	a1d 25	. 2 |- ( -. B e. dom F -> ( Smo F -> ( F ` B ) e. On ) )
10	5, 9	pm2.61i 176	1 |- ( Smo F -> ( F ` B ) e. On )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 1990   A.wral 2912   (/)c0 3915   dom cdm 5114   Ord word 5722   Oncon0 5723   -->wf 5884   ` cfv 5888   Smo wsmo 7442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-id 5024  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-ord 5726  df-on 5727  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-smo 7443

This theorem is referenced by:  smo11  7461  smoord  7462  smoword  7463  smogt  7464