Theorem smoiso 6463

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 5943	. . . 4 |- ( F Isom _E , _E ( A , B ) -> F : A -1-1-onto-> B )
2		f1of 5580	. . . 4 |- ( F : A -1-1-onto-> B -> F : A --> B )
3	1, 2	syl 14	. . 3 |- ( F Isom _E , _E ( A , B ) -> F : A --> B )
4		ffdm 5502	. . . . . 6 |- ( F : A --> B -> ( F : dom F --> B /\ dom F C_ A ) )
5	4	simpld 112	. . . . 5 |- ( F : A --> B -> F : dom F --> B )
6		fss 5491	. . . . 5 |- ( ( F : dom F --> B /\ B C_ On ) -> F : dom F --> On )
7	5, 6	sylan 283	. . . 4 |- ( ( F : A --> B /\ B C_ On ) -> F : dom F --> On )
8	7	3adant2 1040	. . 3 |- ( ( F : A --> B /\ Ord A /\ B C_ On ) -> F : dom F --> On )
9	3, 8	syl3an1 1304	. 2 |- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ B C_ On ) -> F : dom F --> On )
10		fdm 5485	. . . . . 6 |- ( F : A --> B -> dom F = A )
11	10	eqcomd 2235	. . . . 5 |- ( F : A --> B -> A = dom F )
12		ordeq 4467	. . . . 5 |- ( A = dom F -> ( Ord A <-> Ord dom F ) )
13	1, 2, 11, 12	4syl 18	. . . 4 |- ( F Isom _E , _E ( A , B ) -> ( Ord A <-> Ord dom F ) )
14	13	biimpa 296	. . 3 |- ( ( F Isom _E , _E ( A , B ) /\ Ord A ) -> Ord dom F )
15	14	3adant3 1041	. 2 |- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ B C_ On ) -> Ord dom F )
16	10	eleq2d 2299	. . . . . . 7 |- ( F : A --> B -> ( x e. dom F <-> x e. A ) )
17	10	eleq2d 2299	. . . . . . 7 |- ( F : A --> B -> ( y e. dom F <-> y e. A ) )
18	16, 17	anbi12d 473	. . . . . 6 |- ( F : A --> B -> ( ( x e. dom F /\ y e. dom F ) <-> ( x e. A /\ y e. A ) ) )
19	1, 2, 18	3syl 17	. . . . 5 |- ( F Isom _E , _E ( A , B ) -> ( ( x e. dom F /\ y e. dom F ) <-> ( x e. A /\ y e. A ) ) )
20		epel 4387	. . . . . . . . 9 |- ( x _E y <-> x e. y )
21		isorel 5944	. . . . . . . . 9 |- ( ( F Isom _E , _E ( A , B ) /\ ( x e. A /\ y e. A ) ) -> ( x _E y <-> ( F ` x ) _E ( F ` y ) ) )
22	20, 21	bitr3id 194	. . . . . . . 8 |- ( ( F Isom _E , _E ( A , B ) /\ ( x e. A /\ y e. A ) ) -> ( x e. y <-> ( F ` x ) _E ( F ` y ) ) )
23		ffn 5479	. . . . . . . . . . 11 |- ( F : A --> B -> F Fn A )
24	3, 23	syl 14	. . . . . . . . . 10 |- ( F Isom _E , _E ( A , B ) -> F Fn A )
25	24	adantr 276	. . . . . . . . 9 |- ( ( F Isom _E , _E ( A , B ) /\ ( x e. A /\ y e. A ) ) -> F Fn A )
26		simprr 531	. . . . . . . . 9 |- ( ( F Isom _E , _E ( A , B ) /\ ( x e. A /\ y e. A ) ) -> y e. A )
27		funfvex 5652	. . . . . . . . . . 11 |- ( ( Fun F /\ y e. dom F ) -> ( F ` y ) e. _V )
28	27	funfni 5429	. . . . . . . . . 10 |- ( ( F Fn A /\ y e. A ) -> ( F ` y ) e. _V )
29		epelg 4385	. . . . . . . . . 10 |- ( ( F ` y ) e. _V -> ( ( F ` x ) _E ( F ` y ) <-> ( F ` x ) e. ( F ` y ) ) )
30	28, 29	syl 14	. . . . . . . . 9 |- ( ( F Fn A /\ y e. A ) -> ( ( F ` x ) _E ( F ` y ) <-> ( F ` x ) e. ( F ` y ) ) )
31	25, 26, 30	syl2anc 411	. . . . . . . 8 |- ( ( F Isom _E , _E ( A , B ) /\ ( x e. A /\ y e. A ) ) -> ( ( F ` x ) _E ( F ` y ) <-> ( F ` x ) e. ( F ` y ) ) )
32	22, 31	bitrd 188	. . . . . . 7 |- ( ( F Isom _E , _E ( A , B ) /\ ( x e. A /\ y e. A ) ) -> ( x e. y <-> ( F ` x ) e. ( F ` y ) ) )
33	32	biimpd 144	. . . . . 6 |- ( ( F Isom _E , _E ( A , B ) /\ ( x e. A /\ y e. A ) ) -> ( x e. y -> ( F ` x ) e. ( F ` y ) ) )
34	33	ex 115	. . . . 5 |- ( F Isom _E , _E ( A , B ) -> ( ( x e. A /\ y e. A ) -> ( x e. y -> ( F ` x ) e. ( F ` y ) ) ) )
35	19, 34	sylbid 150	. . . 4 |- ( F Isom _E , _E ( A , B ) -> ( ( x e. dom F /\ y e. dom F ) -> ( x e. y -> ( F ` x ) e. ( F ` y ) ) ) )
36	35	ralrimivv 2611	. . 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 ) ) )
37	36	3ad2ant1 1042	. 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 ) ) )
38		df-smo 6447	. 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 ) ) ) )
39	9, 15, 37, 38	syl3anbrc 1205	1 |- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ B C_ On ) -> Smo F )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508   _Vcvv 2800   C_ wss 3198   class class class wbr 4086   _E cep 4382   Ord word 4457   Oncon0 4458   dom cdm 4723   Fn wfn 5319   -->wf 5320   -1-1-onto->wf1o 5323   ` cfv 5324   Isom wiso 5325   Smo wsmo 6446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-sbc 3030  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-tr 4186  df-eprel 4384  df-id 4388  df-iord 4461  df-cnv 4731  df-co 4732  df-dm 4733  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-f1o 5331  df-fv 5332  df-isom 5333  df-smo 6447

This theorem is referenced by: (None)