Theorem smoiso2 7085

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 5791	. . . . . . 7 |- ( F : A -onto-> B -> F : A --> B )
2		smo11 7080	. . . . . . 7 |- ( ( F : A --> B /\ Smo F ) -> F : A -1-1-> B )
3	1, 2	sylan 474	. . . . . 6 |- ( ( F : A -onto-> B /\ Smo F ) -> F : A -1-1-> B )
4		simpl 459	. . . . . 6 |- ( ( F : A -onto-> B /\ Smo F ) -> F : A -onto-> B )
5		df-f1o 5588	. . . . . 6 |- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ F : A -onto-> B ) )
6	3, 4, 5	sylanbrc 669	. . . . 5 |- ( ( F : A -onto-> B /\ Smo F ) -> F : A -1-1-onto-> B )
7	6	adantl 468	. . . 4 |- ( ( ( Ord A /\ B C_ On ) /\ ( F : A -onto-> B /\ Smo F ) ) -> F : A -1-1-onto-> B )
8		fofn 5793	. . . . . 6 |- ( F : A -onto-> B -> F Fn A )
9		smoord 7081	. . . . . . . 8 |- ( ( ( F Fn A /\ Smo F ) /\ ( x e. A /\ y e. A ) ) -> ( x e. y <-> ( F ` x ) e. ( F ` y ) ) )
10		epel 4747	. . . . . . . 8 |- ( x _E y <-> x e. y )
11		fvex 5873	. . . . . . . . 9 |- ( F ` y ) e. _V
12	11	epelc 4746	. . . . . . . 8 |- ( ( F ` x ) _E ( F ` y ) <-> ( F ` x ) e. ( F ` y ) )
13	9, 10, 12	3bitr4g 292	. . . . . . 7 |- ( ( ( F Fn A /\ Smo F ) /\ ( x e. A /\ y e. A ) ) -> ( x _E y <-> ( F ` x ) _E ( F ` y ) ) )
14	13	ralrimivva 2808	. . . . . 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 474	. . . . 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 468	. . . 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 5590	. . . 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 669	. . 3 |- ( ( ( Ord A /\ B C_ On ) /\ ( F : A -onto-> B /\ Smo F ) ) -> F Isom _E , _E ( A , B ) )
19	18	ex 436	. 2 |- ( ( Ord A /\ B C_ On ) -> ( ( F : A -onto-> B /\ Smo F ) -> F Isom _E , _E ( A , B ) ) )
20		isof1o 6214	. . . . . . 7 |- ( F Isom _E , _E ( A , B ) -> F : A -1-1-onto-> B )
21		f1ofo 5819	. . . . . . 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 1028	. . . . 5 |- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ B C_ On ) -> F : A -onto-> B )
24		smoiso 7078	. . . . 5 |- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ B C_ On ) -> Smo F )
25	23, 24	jca 535	. . . 4 |- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ B C_ On ) -> ( F : A -onto-> B /\ Smo F ) )
26	25	3expib 1210	. . 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 194	1 |- ( ( Ord A /\ B C_ On ) -> ( ( F : A -onto-> B /\ Smo F ) <-> F Isom _E , _E ( A , B ) ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 984   e. wcel 1886   A.wral 2736   C_ wss 3403   class class class wbr 4401   _E cep 4742   Ord word 5421   Oncon0 5422   Fn wfn 5576   -->wf 5577   -1-1->wf1 5578   -onto->wfo 5579   -1-1-onto->wf1o 5580   ` cfv 5581   Isom wiso 5582   Smo wsmo 7061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-ord 5425  df-on 5426  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-smo 7062

This theorem is referenced by:  oismo  8052  cofsmo  8696