Theorem smoiso2 7041

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 5795	. . . . . . 7 |- ( F : A -onto-> B -> F : A --> B )
2		smo11 7036	. . . . . . 7 |- ( ( F : A --> B /\ Smo F ) -> F : A -1-1-> B )
3	1, 2	sylan 471	. . . . . 6 |- ( ( F : A -onto-> B /\ Smo F ) -> F : A -1-1-> B )
4		simpl 457	. . . . . 6 |- ( ( F : A -onto-> B /\ Smo F ) -> F : A -onto-> B )
5		df-f1o 5595	. . . . . 6 |- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ F : A -onto-> B ) )
6	3, 4, 5	sylanbrc 664	. . . . 5 |- ( ( F : A -onto-> B /\ Smo F ) -> F : A -1-1-onto-> B )
7	6	adantl 466	. . . 4 |- ( ( ( Ord A /\ B C_ On ) /\ ( F : A -onto-> B /\ Smo F ) ) -> F : A -1-1-onto-> B )
8		fofn 5797	. . . . . 6 |- ( F : A -onto-> B -> F Fn A )
9		smoord 7037	. . . . . . . 8 |- ( ( ( F Fn A /\ Smo F ) /\ ( x e. A /\ y e. A ) ) -> ( x e. y <-> ( F ` x ) e. ( F ` y ) ) )
10		epel 4794	. . . . . . . 8 |- ( x _E y <-> x e. y )
11		fvex 5876	. . . . . . . . 9 |- ( F ` y ) e. _V
12	11	epelc 4793	. . . . . . . 8 |- ( ( F ` x ) _E ( F ` y ) <-> ( F ` x ) e. ( F ` y ) )
13	9, 10, 12	3bitr4g 288	. . . . . . 7 |- ( ( ( F Fn A /\ Smo F ) /\ ( x e. A /\ y e. A ) ) -> ( x _E y <-> ( F ` x ) _E ( F ` y ) ) )
14	13	ralrimivva 2885	. . . . . 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 471	. . . . 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 466	. . . 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 5597	. . . 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 664	. . 3 |- ( ( ( Ord A /\ B C_ On ) /\ ( F : A -onto-> B /\ Smo F ) ) -> F Isom _E , _E ( A , B ) )
19	18	ex 434	. 2 |- ( ( Ord A /\ B C_ On ) -> ( ( F : A -onto-> B /\ Smo F ) -> F Isom _E , _E ( A , B ) ) )
20		isof1o 6210	. . . . . . 7 |- ( F Isom _E , _E ( A , B ) -> F : A -1-1-onto-> B )
21		f1ofo 5823	. . . . . . 7 |- ( F : A -1-1-onto-> B -> F : A -onto-> B )
22	20, 21	syl 16	. . . . . 6 |- ( F Isom _E , _E ( A , B ) -> F : A -onto-> B )
23	22	3ad2ant1 1017	. . . . 5 |- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ B C_ On ) -> F : A -onto-> B )
24		smoiso 7034	. . . . 5 |- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ B C_ On ) -> Smo F )
25	23, 24	jca 532	. . . 4 |- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ B C_ On ) -> ( F : A -onto-> B /\ Smo F ) )
26	25	3expib 1199	. . 3 |- ( F Isom _E , _E ( A , B ) -> ( ( Ord A /\ B C_ On ) -> ( F : A -onto-> B /\ Smo F ) ) )
27	26	com12 31	. 2 |- ( ( Ord A /\ B C_ On ) -> ( F Isom _E , _E ( A , B ) -> ( F : A -onto-> B /\ Smo F ) ) )
28	19, 27	impbid 191	1 |- ( ( Ord A /\ B C_ On ) -> ( ( F : A -onto-> B /\ Smo F ) <-> F Isom _E , _E ( A , B ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   e. wcel 1767   A.wral 2814   C_ wss 3476   class class class wbr 4447   _E cep 4789   Ord word 4877   Oncon0 4878   Fn wfn 5583   -->wf 5584   -1-1->wf1 5585   -onto->wfo 5586   -1-1-onto->wf1o 5587   ` cfv 5588   Isom wiso 5589   Smo wsmo 7017

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-smo 7018

This theorem is referenced by:  oismo  7966  cofsmo  8650