Theorem smoiso 6535

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 5982	. . . 4 |- ( F Isom _E , _E ( A , B ) -> F : A -1-1-onto-> B )
2		f1of 5616	. . . 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 5535	. . . . . 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 5523	. . . . 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 1043	. . 3 |- ( ( F : A --> B /\ Ord A /\ B C_ On ) -> F : dom F --> On )
9	3, 8	syl3an1 1307	. 2 |- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ B C_ On ) -> F : dom F --> On )
10		fdm 5516	. . . . . 6 |- ( F : A --> B -> dom F = A )
11	10	eqcomd 2240	. . . . 5 |- ( F : A --> B -> A = dom F )
12		ordeq 4495	. . . . 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 1044	. 2 |- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ B C_ On ) -> Ord dom F )
16	10	eleq2d 2304	. . . . . . 7 |- ( F : A --> B -> ( x e. dom F <-> x e. A ) )
17	10	eleq2d 2304	. . . . . . 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 4415	. . . . . . . . 9 |- ( x _E y <-> x e. y )
21		isorel 5983	. . . . . . . . 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 5510	. . . . . . . . . . 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 533	. . . . . . . . 9 |- ( ( F Isom _E , _E ( A , B ) /\ ( x e. A /\ y e. A ) ) -> y e. A )
27		funfvex 5689	. . . . . . . . . . 11 |- ( ( Fun F /\ y e. dom F ) -> ( F ` y ) e. _V )
28	27	funfni 5460	. . . . . . . . . 10 |- ( ( F Fn A /\ y e. A ) -> ( F ` y ) e. _V )
29		epelg 4413	. . . . . . . . . 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 2625	. . 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 1045	. 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 6519	. 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 1208	1 |- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ B C_ On ) -> Smo F )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   A.wral 2522   _Vcvv 2815   C_ wss 3213   class class class wbr 4111   _E cep 4410   Ord word 4485   Oncon0 4486   dom cdm 4751   Fn wfn 5349   -->wf 5350   -1-1-onto->wf1o 5353   ` cfv 5354   Isom wiso 5355   Smo wsmo 6518

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-tr 4211  df-eprel 4412  df-id 4416  df-iord 4489  df-cnv 4759  df-co 4760  df-dm 4761  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-f1o 5361  df-fv 5362  df-isom 5363  df-smo 6519

This theorem is referenced by: (None)