Theorem smoiso 6249

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 5757	. . . 4 |- ( F Isom _E , _E ( A , B ) -> F : A -1-1-onto-> B )
2		f1of 5414	. . . 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 5340	. . . . . 6 |- ( F : A --> B -> ( F : dom F --> B /\ dom F C_ A ) )
5	4	simpld 111	. . . . 5 |- ( F : A --> B -> F : dom F --> B )
6		fss 5331	. . . . 5 |- ( ( F : dom F --> B /\ B C_ On ) -> F : dom F --> On )
7	5, 6	sylan 281	. . . 4 |- ( ( F : A --> B /\ B C_ On ) -> F : dom F --> On )
8	7	3adant2 1001	. . 3 |- ( ( F : A --> B /\ Ord A /\ B C_ On ) -> F : dom F --> On )
9	3, 8	syl3an1 1253	. 2 |- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ B C_ On ) -> F : dom F --> On )
10		fdm 5325	. . . . . 6 |- ( F : A --> B -> dom F = A )
11	10	eqcomd 2163	. . . . 5 |- ( F : A --> B -> A = dom F )
12		ordeq 4332	. . . . 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 294	. . 3 |- ( ( F Isom _E , _E ( A , B ) /\ Ord A ) -> Ord dom F )
15	14	3adant3 1002	. 2 |- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ B C_ On ) -> Ord dom F )
16	10	eleq2d 2227	. . . . . . 7 |- ( F : A --> B -> ( x e. dom F <-> x e. A ) )
17	10	eleq2d 2227	. . . . . . 7 |- ( F : A --> B -> ( y e. dom F <-> y e. A ) )
18	16, 17	anbi12d 465	. . . . . 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 4252	. . . . . . . . 9 |- ( x _E y <-> x e. y )
21		isorel 5758	. . . . . . . . 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 193	. . . . . . . 8 |- ( ( F Isom _E , _E ( A , B ) /\ ( x e. A /\ y e. A ) ) -> ( x e. y <-> ( F ` x ) _E ( F ` y ) ) )
23		ffn 5319	. . . . . . . . . . 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 274	. . . . . . . . 9 |- ( ( F Isom _E , _E ( A , B ) /\ ( x e. A /\ y e. A ) ) -> F Fn A )
26		simprr 522	. . . . . . . . 9 |- ( ( F Isom _E , _E ( A , B ) /\ ( x e. A /\ y e. A ) ) -> y e. A )
27		funfvex 5485	. . . . . . . . . . 11 |- ( ( Fun F /\ y e. dom F ) -> ( F ` y ) e. _V )
28	27	funfni 5270	. . . . . . . . . 10 |- ( ( F Fn A /\ y e. A ) -> ( F ` y ) e. _V )
29		epelg 4250	. . . . . . . . . 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 409	. . . . . . . 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 187	. . . . . . 7 |- ( ( F Isom _E , _E ( A , B ) /\ ( x e. A /\ y e. A ) ) -> ( x e. y <-> ( F ` x ) e. ( F ` y ) ) )
33	32	biimpd 143	. . . . . 6 |- ( ( F Isom _E , _E ( A , B ) /\ ( x e. A /\ y e. A ) ) -> ( x e. y -> ( F ` x ) e. ( F ` y ) ) )
34	33	ex 114	. . . . 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 149	. . . 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 2538	. . 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 1003	. 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 6233	. 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 1166	1 |- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ B C_ On ) -> Smo F )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 963   = wceq 1335   e. wcel 2128   A.wral 2435   _Vcvv 2712   C_ wss 3102   class class class wbr 3965   _E cep 4247   Ord word 4322   Oncon0 4323   dom cdm 4586   Fn wfn 5165   -->wf 5166   -1-1-onto->wf1o 5169   ` cfv 5170   Isom wiso 5171   Smo wsmo 6232

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-tr 4063  df-eprel 4249  df-id 4253  df-iord 4326  df-cnv 4594  df-co 4595  df-dm 4596  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-f1o 5177  df-fv 5178  df-isom 5179  df-smo 6233

This theorem is referenced by: (None)