Theorem smoiso 6411

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 5899	. . . 4 |- ( F Isom _E , _E ( A , B ) -> F : A -1-1-onto-> B )
2		f1of 5544	. . . 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 5466	. . . . . 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 5457	. . . . 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 1019	. . 3 |- ( ( F : A --> B /\ Ord A /\ B C_ On ) -> F : dom F --> On )
9	3, 8	syl3an1 1283	. 2 |- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ B C_ On ) -> F : dom F --> On )
10		fdm 5451	. . . . . 6 |- ( F : A --> B -> dom F = A )
11	10	eqcomd 2213	. . . . 5 |- ( F : A --> B -> A = dom F )
12		ordeq 4437	. . . . 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 1020	. 2 |- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ B C_ On ) -> Ord dom F )
16	10	eleq2d 2277	. . . . . . 7 |- ( F : A --> B -> ( x e. dom F <-> x e. A ) )
17	10	eleq2d 2277	. . . . . . 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 4357	. . . . . . . . 9 |- ( x _E y <-> x e. y )
21		isorel 5900	. . . . . . . . 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 5445	. . . . . . . . . . 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 5616	. . . . . . . . . . 11 |- ( ( Fun F /\ y e. dom F ) -> ( F ` y ) e. _V )
28	27	funfni 5395	. . . . . . . . . 10 |- ( ( F Fn A /\ y e. A ) -> ( F ` y ) e. _V )
29		epelg 4355	. . . . . . . . . 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 2589	. . 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 1021	. 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 6395	. 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 1184	1 |- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ B C_ On ) -> Smo F )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2178   A.wral 2486   _Vcvv 2776   C_ wss 3174   class class class wbr 4059   _E cep 4352   Ord word 4427   Oncon0 4428   dom cdm 4693   Fn wfn 5285   -->wf 5286   -1-1-onto->wf1o 5289   ` cfv 5290   Isom wiso 5291   Smo wsmo 6394

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-tr 4159  df-eprel 4354  df-id 4358  df-iord 4431  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-f1o 5297  df-fv 5298  df-isom 5299  df-smo 6395

This theorem is referenced by: (None)