Theorem smorndom 7465

Description: The range of a strictly monotone ordinal function dominates the domain. (Contributed by Mario Carneiro, 13-Mar-2013.)

Assertion

Ref	Expression
smorndom	|- ( ( F : A --> B /\ Smo F /\ Ord B ) -> A C_ B )

Proof of Theorem smorndom

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 1064	. . . . . . 7 |- ( ( ( F : A --> B /\ Smo F /\ Ord B ) /\ x e. A ) -> F : A --> B )
2		ffn 6045	. . . . . . 7 |- ( F : A --> B -> F Fn A )
3	1, 2	syl 17	. . . . . 6 |- ( ( ( F : A --> B /\ Smo F /\ Ord B ) /\ x e. A ) -> F Fn A )
4		simpl2 1065	. . . . . 6 |- ( ( ( F : A --> B /\ Smo F /\ Ord B ) /\ x e. A ) -> Smo F )
5		smodm2 7452	. . . . . 6 |- ( ( F Fn A /\ Smo F ) -> Ord A )
6	3, 4, 5	syl2anc 693	. . . . 5 |- ( ( ( F : A --> B /\ Smo F /\ Ord B ) /\ x e. A ) -> Ord A )
7		ordelord 5745	. . . . 5 |- ( ( Ord A /\ x e. A ) -> Ord x )
8	6, 7	sylancom 701	. . . 4 |- ( ( ( F : A --> B /\ Smo F /\ Ord B ) /\ x e. A ) -> Ord x )
9		simpl3 1066	. . . 4 |- ( ( ( F : A --> B /\ Smo F /\ Ord B ) /\ x e. A ) -> Ord B )
10		simpr 477	. . . . 5 |- ( ( ( F : A --> B /\ Smo F /\ Ord B ) /\ x e. A ) -> x e. A )
11		smogt 7464	. . . . 5 |- ( ( F Fn A /\ Smo F /\ x e. A ) -> x C_ ( F ` x ) )
12	3, 4, 10, 11	syl3anc 1326	. . . 4 |- ( ( ( F : A --> B /\ Smo F /\ Ord B ) /\ x e. A ) -> x C_ ( F ` x ) )
13		ffvelrn 6357	. . . . 5 |- ( ( F : A --> B /\ x e. A ) -> ( F ` x ) e. B )
14	13	3ad2antl1 1223	. . . 4 |- ( ( ( F : A --> B /\ Smo F /\ Ord B ) /\ x e. A ) -> ( F ` x ) e. B )
15		ordtr2 5768	. . . . 5 |- ( ( Ord x /\ Ord B ) -> ( ( x C_ ( F ` x ) /\ ( F ` x ) e. B ) -> x e. B ) )
16	15	imp 445	. . . 4 |- ( ( ( Ord x /\ Ord B ) /\ ( x C_ ( F ` x ) /\ ( F ` x ) e. B ) ) -> x e. B )
17	8, 9, 12, 14, 16	syl22anc 1327	. . 3 |- ( ( ( F : A --> B /\ Smo F /\ Ord B ) /\ x e. A ) -> x e. B )
18	17	ex 450	. 2 |- ( ( F : A --> B /\ Smo F /\ Ord B ) -> ( x e. A -> x e. B ) )
19	18	ssrdv 3609	1 |- ( ( F : A --> B /\ Smo F /\ Ord B ) -> A C_ B )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   e. wcel 1990   C_ wss 3574   Ord word 5722   Fn wfn 5883   -->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  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-id 5024  df-eprel 5029  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:  cofsmo  9091  hsmexlem1  9248