Theorem smoiso2 7106

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 5806	. . . . . . 7 |- ( F : A -onto-> B -> F : A --> B )
2		smo11 7101	. . . . . . 7 |- ( ( F : A --> B /\ Smo F ) -> F : A -1-1-> B )
3	1, 2	sylan 479	. . . . . 6 |- ( ( F : A -onto-> B /\ Smo F ) -> F : A -1-1-> B )
4		simpl 464	. . . . . 6 |- ( ( F : A -onto-> B /\ Smo F ) -> F : A -onto-> B )
5		df-f1o 5596	. . . . . 6 |- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ F : A -onto-> B ) )
6	3, 4, 5	sylanbrc 677	. . . . 5 |- ( ( F : A -onto-> B /\ Smo F ) -> F : A -1-1-onto-> B )
7	6	adantl 473	. . . 4 |- ( ( ( Ord A /\ B C_ On ) /\ ( F : A -onto-> B /\ Smo F ) ) -> F : A -1-1-onto-> B )
8		fofn 5808	. . . . . 6 |- ( F : A -onto-> B -> F Fn A )
9		smoord 7102	. . . . . . . 8 |- ( ( ( F Fn A /\ Smo F ) /\ ( x e. A /\ y e. A ) ) -> ( x e. y <-> ( F ` x ) e. ( F ` y ) ) )
10		epel 4753	. . . . . . . 8 |- ( x _E y <-> x e. y )
11		fvex 5889	. . . . . . . . 9 |- ( F ` y ) e. _V
12	11	epelc 4752	. . . . . . . 8 |- ( ( F ` x ) _E ( F ` y ) <-> ( F ` x ) e. ( F ` y ) )
13	9, 10, 12	3bitr4g 296	. . . . . . 7 |- ( ( ( F Fn A /\ Smo F ) /\ ( x e. A /\ y e. A ) ) -> ( x _E y <-> ( F ` x ) _E ( F ` y ) ) )
14	13	ralrimivva 2814	. . . . . 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 479	. . . . 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 473	. . . 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 5598	. . . 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 677	. . 3 |- ( ( ( Ord A /\ B C_ On ) /\ ( F : A -onto-> B /\ Smo F ) ) -> F Isom _E , _E ( A , B ) )
19	18	ex 441	. 2 |- ( ( Ord A /\ B C_ On ) -> ( ( F : A -onto-> B /\ Smo F ) -> F Isom _E , _E ( A , B ) ) )
20		isof1o 6234	. . . . . . 7 |- ( F Isom _E , _E ( A , B ) -> F : A -1-1-onto-> B )
21		f1ofo 5835	. . . . . . 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 1051	. . . . 5 |- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ B C_ On ) -> F : A -onto-> B )
24		smoiso 7099	. . . . 5 |- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ B C_ On ) -> Smo F )
25	23, 24	jca 541	. . . 4 |- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ B C_ On ) -> ( F : A -onto-> B /\ Smo F ) )
26	25	3expib 1234	. . 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 195	1 |- ( ( Ord A /\ B C_ On ) -> ( ( F : A -onto-> B /\ Smo F ) <-> F Isom _E , _E ( A , B ) ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   /\ w3a 1007   e. wcel 1904   A.wral 2756   C_ wss 3390   class class class wbr 4395   _E cep 4748   Ord word 5429   Oncon0 5430   Fn wfn 5584   -->wf 5585   -1-1->wf1 5586   -onto->wfo 5587   -1-1-onto->wf1o 5588   ` cfv 5589   Isom wiso 5590   Smo wsmo 7082

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-ord 5433  df-on 5434  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-smo 7083

This theorem is referenced by:  oismo  8073  cofsmo  8717