Theorem smoiso 7459

Description: If F is an isomorphism from an ordinal A onto B, which is a subset of the ordinals, then F is a strictly monotonic function. Exercise 3 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 24-Nov-2011.)

Assertion

Ref	Expression
smoiso	|- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ B C_ On ) -> Smo F )

Proof of Theorem smoiso

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

Step	Hyp	Ref	Expression
1		isof1o 6573	. . . 4 |- ( F Isom _E , _E ( A , B ) -> F : A -1-1-onto-> B )
2		f1of 6137	. . . 4 |- ( F : A -1-1-onto-> B -> F : A --> B )
3	1, 2	syl 17	. . 3 |- ( F Isom _E , _E ( A , B ) -> F : A --> B )
4		ffdm 6062	. . . . . 6 |- ( F : A --> B -> ( F : dom F --> B /\ dom F C_ A ) )
5	4	simpld 475	. . . . 5 |- ( F : A --> B -> F : dom F --> B )
6		fss 6056	. . . . 5 |- ( ( F : dom F --> B /\ B C_ On ) -> F : dom F --> On )
7	5, 6	sylan 488	. . . 4 |- ( ( F : A --> B /\ B C_ On ) -> F : dom F --> On )
8	7	3adant2 1080	. . 3 |- ( ( F : A --> B /\ Ord A /\ B C_ On ) -> F : dom F --> On )
9	3, 8	syl3an1 1359	. 2 |- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ B C_ On ) -> F : dom F --> On )
10		fdm 6051	. . . . . 6 |- ( F : A --> B -> dom F = A )
11	10	eqcomd 2628	. . . . 5 |- ( F : A --> B -> A = dom F )
12		ordeq 5730	. . . . 5 |- ( A = dom F -> ( Ord A <-> Ord dom F ) )
13	1, 2, 11, 12	4syl 19	. . . 4 |- ( F Isom _E , _E ( A , B ) -> ( Ord A <-> Ord dom F ) )
14	13	biimpa 501	. . 3 |- ( ( F Isom _E , _E ( A , B ) /\ Ord A ) -> Ord dom F )
15	14	3adant3 1081	. 2 |- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ B C_ On ) -> Ord dom F )
16	10	eleq2d 2687	. . . . . . 7 |- ( F : A --> B -> ( x e. dom F <-> x e. A ) )
17	10	eleq2d 2687	. . . . . . 7 |- ( F : A --> B -> ( y e. dom F <-> y e. A ) )
18	16, 17	anbi12d 747	. . . . . 6 |- ( F : A --> B -> ( ( x e. dom F /\ y e. dom F ) <-> ( x e. A /\ y e. A ) ) )
19	1, 2, 18	3syl 18	. . . . 5 |- ( F Isom _E , _E ( A , B ) -> ( ( x e. dom F /\ y e. dom F ) <-> ( x e. A /\ y e. A ) ) )
20		isorel 6576	. . . . . . . 8 |- ( ( F Isom _E , _E ( A , B ) /\ ( x e. A /\ y e. A ) ) -> ( x _E y <-> ( F ` x ) _E ( F ` y ) ) )
21		epel 5032	. . . . . . . 8 |- ( x _E y <-> x e. y )
22		fvex 6201	. . . . . . . . 9 |- ( F ` y ) e. _V
23	22	epelc 5031	. . . . . . . 8 |- ( ( F ` x ) _E ( F ` y ) <-> ( F ` x ) e. ( F ` y ) )
24	20, 21, 23	3bitr3g 302	. . . . . . 7 |- ( ( F Isom _E , _E ( A , B ) /\ ( x e. A /\ y e. A ) ) -> ( x e. y <-> ( F ` x ) e. ( F ` y ) ) )
25	24	biimpd 219	. . . . . 6 |- ( ( F Isom _E , _E ( A , B ) /\ ( x e. A /\ y e. A ) ) -> ( x e. y -> ( F ` x ) e. ( F ` y ) ) )
26	25	ex 450	. . . . 5 |- ( F Isom _E , _E ( A , B ) -> ( ( x e. A /\ y e. A ) -> ( x e. y -> ( F ` x ) e. ( F ` y ) ) ) )
27	19, 26	sylbid 230	. . . 4 |- ( F Isom _E , _E ( A , B ) -> ( ( x e. dom F /\ y e. dom F ) -> ( x e. y -> ( F ` x ) e. ( F ` y ) ) ) )
28	27	ralrimivv 2970	. . 3 |- ( F Isom _E , _E ( A , B ) -> A. x e. dom F A. y e. dom F ( x e. y -> ( F ` x ) e. ( F ` y ) ) )
29	28	3ad2ant1 1082	. 2 |- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ B C_ On ) -> A. x e. dom F A. y e. dom F ( x e. y -> ( F ` x ) e. ( F ` y ) ) )
30		df-smo 7443	. 2 |- ( Smo F <-> ( F : dom F --> On /\ Ord dom F /\ A. x e. dom F A. y e. dom F ( x e. y -> ( F ` x ) e. ( F ` y ) ) ) )
31	9, 15, 29, 30	syl3anbrc 1246	1 |- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ B C_ On ) -> Smo F )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   class class class wbr 4653   _E cep 5028   dom cdm 5114   Ord word 5722   Oncon0 5723   -->wf 5884   -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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-iota 5851  df-fn 5891  df-f 5892  df-f1 5893  df-f1o 5895  df-fv 5896  df-isom 5897  df-smo 7443

This theorem is referenced by:  smoiso2  7466