Theorem smoiso2 7075

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 5780	. . . . . . 7 |- ( F : A -onto-> B -> F : A --> B )
2		smo11 7070	. . . . . . 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 5578	. . . . . 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 5782	. . . . . 6 |- ( F : A -onto-> B -> F Fn A )
9		smoord 7071	. . . . . . . 8 |- ( ( ( F Fn A /\ Smo F ) /\ ( x e. A /\ y e. A ) ) -> ( x e. y <-> ( F ` x ) e. ( F ` y ) ) )
10		epel 4739	. . . . . . . 8 |- ( x _E y <-> x e. y )
11		fvex 5861	. . . . . . . . 9 |- ( F ` y ) e. _V
12	11	epelc 4738	. . . . . . . 8 |- ( ( F ` x ) _E ( F ` y ) <-> ( F ` x ) e. ( F ` y ) )
13	9, 10, 12	3bitr4g 290	. . . . . . 7 |- ( ( ( F Fn A /\ Smo F ) /\ ( x e. A /\ y e. A ) ) -> ( x _E y <-> ( F ` x ) _E ( F ` y ) ) )
14	13	ralrimivva 2827	. . . . . 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 5580	. . . 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 6206	. . . . . . 7 |- ( F Isom _E , _E ( A , B ) -> F : A -1-1-onto-> B )
21		f1ofo 5808	. . . . . . 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 1020	. . . . 5 |- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ B C_ On ) -> F : A -onto-> B )
24		smoiso 7068	. . . . 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 1202	. . 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 192	1 |- ( ( Ord A /\ B C_ On ) -> ( ( F : A -onto-> B /\ Smo F ) <-> F Isom _E , _E ( A , B ) ) )

Syntax hints:   -> wi 4   <-> wb 186   /\ wa 369   /\ w3a 976   e. wcel 1844   A.wral 2756   C_ wss 3416   class class class wbr 4397   _E cep 4734   Ord word 5411   Oncon0 5412   Fn wfn 5566   -->wf 5567   -1-1->wf1 5568   -onto->wfo 5569   -1-1-onto->wf1o 5570   ` cfv 5571   Isom wiso 5572   Smo wsmo 7051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632

This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-opab 4456  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-ord 5415  df-on 5416  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-isom 5580  df-smo 7052

This theorem is referenced by:  oismo  8001  cofsmo  8683