Theorem smoword 7463

Description: A strictly monotone ordinal function preserves weak ordering. (Contributed by Mario Carneiro, 4-Mar-2013.)

Assertion

Ref	Expression
smoword	|- ( ( ( F Fn A /\ Smo F ) /\ ( C e. A /\ D e. A ) ) -> ( C C_ D <-> ( F ` C ) C_ ( F ` D ) ) )

Proof of Theorem smoword

Step	Hyp	Ref	Expression
1		smoord 7462	. . . 4 |- ( ( ( F Fn A /\ Smo F ) /\ ( D e. A /\ C e. A ) ) -> ( D e. C <-> ( F ` D ) e. ( F ` C ) ) )
2	1	notbid 308	. . 3 |- ( ( ( F Fn A /\ Smo F ) /\ ( D e. A /\ C e. A ) ) -> ( -. D e. C <-> -. ( F ` D ) e. ( F ` C ) ) )
3	2	ancom2s 844	. 2 |- ( ( ( F Fn A /\ Smo F ) /\ ( C e. A /\ D e. A ) ) -> ( -. D e. C <-> -. ( F ` D ) e. ( F ` C ) ) )
4		smodm2 7452	. . . . 5 |- ( ( F Fn A /\ Smo F ) -> Ord A )
5	4	adantr 481	. . . 4 |- ( ( ( F Fn A /\ Smo F ) /\ ( C e. A /\ D e. A ) ) -> Ord A )
6		simprl 794	. . . 4 |- ( ( ( F Fn A /\ Smo F ) /\ ( C e. A /\ D e. A ) ) -> C e. A )
7		ordelord 5745	. . . 4 |- ( ( Ord A /\ C e. A ) -> Ord C )
8	5, 6, 7	syl2anc 693	. . 3 |- ( ( ( F Fn A /\ Smo F ) /\ ( C e. A /\ D e. A ) ) -> Ord C )
9		simprr 796	. . . 4 |- ( ( ( F Fn A /\ Smo F ) /\ ( C e. A /\ D e. A ) ) -> D e. A )
10		ordelord 5745	. . . 4 |- ( ( Ord A /\ D e. A ) -> Ord D )
11	5, 9, 10	syl2anc 693	. . 3 |- ( ( ( F Fn A /\ Smo F ) /\ ( C e. A /\ D e. A ) ) -> Ord D )
12		ordtri1 5756	. . 3 |- ( ( Ord C /\ Ord D ) -> ( C C_ D <-> -. D e. C ) )
13	8, 11, 12	syl2anc 693	. 2 |- ( ( ( F Fn A /\ Smo F ) /\ ( C e. A /\ D e. A ) ) -> ( C C_ D <-> -. D e. C ) )
14		simplr 792	. . . 4 |- ( ( ( F Fn A /\ Smo F ) /\ ( C e. A /\ D e. A ) ) -> Smo F )
15		smofvon2 7453	. . . 4 |- ( Smo F -> ( F ` C ) e. On )
16		eloni 5733	. . . 4 |- ( ( F ` C ) e. On -> Ord ( F ` C ) )
17	14, 15, 16	3syl 18	. . 3 |- ( ( ( F Fn A /\ Smo F ) /\ ( C e. A /\ D e. A ) ) -> Ord ( F ` C ) )
18		smofvon2 7453	. . . 4 |- ( Smo F -> ( F ` D ) e. On )
19		eloni 5733	. . . 4 |- ( ( F ` D ) e. On -> Ord ( F ` D ) )
20	14, 18, 19	3syl 18	. . 3 |- ( ( ( F Fn A /\ Smo F ) /\ ( C e. A /\ D e. A ) ) -> Ord ( F ` D ) )
21		ordtri1 5756	. . 3 |- ( ( Ord ( F ` C ) /\ Ord ( F ` D ) ) -> ( ( F ` C ) C_ ( F ` D ) <-> -. ( F ` D ) e. ( F ` C ) ) )
22	17, 20, 21	syl2anc 693	. 2 |- ( ( ( F Fn A /\ Smo F ) /\ ( C e. A /\ D e. A ) ) -> ( ( F ` C ) C_ ( F ` D ) <-> -. ( F ` D ) e. ( F ` C ) ) )
23	3, 13, 22	3bitr4d 300	1 |- ( ( ( F Fn A /\ Smo F ) /\ ( C e. A /\ D e. A ) ) -> ( C C_ D <-> ( F ` C ) C_ ( F ` D ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   C_ wss 3574   Ord word 5722   Oncon0 5723   Fn wfn 5883   ` 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-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-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:  cfcoflem  9094  coftr  9095