Theorem smoiso2 7466

Description: The strictly monotone ordinal functions are also epsilon isomorphisms of subclasses of On. (Contributed by Mario Carneiro, 20-Mar-2013.)

Assertion

Ref	Expression
smoiso2	|- ( ( Ord A /\ B C_ On ) -> ( ( F : A -onto-> B /\ Smo F ) <-> F Isom _E , _E ( A , B ) ) )

Proof of Theorem smoiso2

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

Step	Hyp	Ref	Expression
1		fof 6115	. . . . . . 7 |- ( F : A -onto-> B -> F : A --> B )
2		smo11 7461	. . . . . . 7 |- ( ( F : A --> B /\ Smo F ) -> F : A -1-1-> B )
3	1, 2	sylan 488	. . . . . 6 |- ( ( F : A -onto-> B /\ Smo F ) -> F : A -1-1-> B )
4		simpl 473	. . . . . 6 |- ( ( F : A -onto-> B /\ Smo F ) -> F : A -onto-> B )
5		df-f1o 5895	. . . . . 6 |- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ F : A -onto-> B ) )
6	3, 4, 5	sylanbrc 698	. . . . 5 |- ( ( F : A -onto-> B /\ Smo F ) -> F : A -1-1-onto-> B )
7	6	adantl 482	. . . 4 |- ( ( ( Ord A /\ B C_ On ) /\ ( F : A -onto-> B /\ Smo F ) ) -> F : A -1-1-onto-> B )
8		fofn 6117	. . . . . 6 |- ( F : A -onto-> B -> F Fn A )
9		smoord 7462	. . . . . . . 8 |- ( ( ( F Fn A /\ Smo F ) /\ ( x e. A /\ y e. A ) ) -> ( x e. y <-> ( F ` x ) e. ( F ` y ) ) )
10		epel 5032	. . . . . . . 8 |- ( x _E y <-> x e. y )
11		fvex 6201	. . . . . . . . 9 |- ( F ` y ) e. _V
12	11	epelc 5031	. . . . . . . 8 |- ( ( F ` x ) _E ( F ` y ) <-> ( F ` x ) e. ( F ` y ) )
13	9, 10, 12	3bitr4g 303	. . . . . . 7 |- ( ( ( F Fn A /\ Smo F ) /\ ( x e. A /\ y e. A ) ) -> ( x _E y <-> ( F ` x ) _E ( F ` y ) ) )
14	13	ralrimivva 2971	. . . . . 6 |- ( ( F Fn A /\ Smo F ) -> A. x e. A A. y e. A ( x _E y <-> ( F ` x ) _E ( F ` y ) ) )
15	8, 14	sylan 488	. . . . 5 |- ( ( F : A -onto-> B /\ Smo F ) -> A. x e. A A. y e. A ( x _E y <-> ( F ` x ) _E ( F ` y ) ) )
16	15	adantl 482	. . . 4 |- ( ( ( Ord A /\ B C_ On ) /\ ( F : A -onto-> B /\ Smo F ) ) -> A. x e. A A. y e. A ( x _E y <-> ( F ` x ) _E ( F ` y ) ) )
17		df-isom 5897	. . . 4 |- ( F Isom _E , _E ( A , B ) <-> ( F : A -1-1-onto-> B /\ A. x e. A A. y e. A ( x _E y <-> ( F ` x ) _E ( F ` y ) ) ) )
18	7, 16, 17	sylanbrc 698	. . 3 |- ( ( ( Ord A /\ B C_ On ) /\ ( F : A -onto-> B /\ Smo F ) ) -> F Isom _E , _E ( A , B ) )
19	18	ex 450	. 2 |- ( ( Ord A /\ B C_ On ) -> ( ( F : A -onto-> B /\ Smo F ) -> F Isom _E , _E ( A , B ) ) )
20		isof1o 6573	. . . . . . 7 |- ( F Isom _E , _E ( A , B ) -> F : A -1-1-onto-> B )
21		f1ofo 6144	. . . . . . 7 |- ( F : A -1-1-onto-> B -> F : A -onto-> B )
22	20, 21	syl 17	. . . . . 6 |- ( F Isom _E , _E ( A , B ) -> F : A -onto-> B )
23	22	3ad2ant1 1082	. . . . 5 |- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ B C_ On ) -> F : A -onto-> B )
24		smoiso 7459	. . . . 5 |- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ B C_ On ) -> Smo F )
25	23, 24	jca 554	. . . 4 |- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ B C_ On ) -> ( F : A -onto-> B /\ Smo F ) )
26	25	3expib 1268	. . 3 |- ( F Isom _E , _E ( A , B ) -> ( ( Ord A /\ B C_ On ) -> ( F : A -onto-> B /\ Smo F ) ) )
27	26	com12 32	. 2 |- ( ( Ord A /\ B C_ On ) -> ( F Isom _E , _E ( A , B ) -> ( F : A -onto-> B /\ Smo F ) ) )
28	19, 27	impbid 202	1 |- ( ( Ord A /\ B C_ On ) -> ( ( F : A -onto-> B /\ Smo F ) <-> F Isom _E , _E ( A , B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   e. wcel 1990   A.wral 2912   C_ wss 3574   class class class wbr 4653   _E cep 5028   Ord word 5722   Oncon0 5723   Fn wfn 5883   -->wf 5884   -1-1->wf1 5885   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888   Isom wiso 5889   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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-smo 7443

This theorem is referenced by:  oismo  8445  cofsmo  9091