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Theorem smoiso 5858
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 ⊆ On) → Smo 𝐹)

Proof of Theorem smoiso
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isof1o 5390 . . . 4 (𝐹 Isom E , E (A, B) → 𝐹:A1-1-ontoB)
2 f1of 5069 . . . 4 (𝐹:A1-1-ontoB𝐹:AB)
31, 2syl 14 . . 3 (𝐹 Isom E , E (A, B) → 𝐹:AB)
4 ffdm 5004 . . . . . 6 (𝐹:AB → (𝐹:dom 𝐹B dom 𝐹A))
54simpld 105 . . . . 5 (𝐹:AB𝐹:dom 𝐹B)
6 fss 4997 . . . . 5 ((𝐹:dom 𝐹B B ⊆ On) → 𝐹:dom 𝐹⟶On)
75, 6sylan 267 . . . 4 ((𝐹:AB B ⊆ On) → 𝐹:dom 𝐹⟶On)
873adant2 922 . . 3 ((𝐹:AB Ord A B ⊆ On) → 𝐹:dom 𝐹⟶On)
93, 8syl3an1 1167 . 2 ((𝐹 Isom E , E (A, B) Ord A B ⊆ On) → 𝐹:dom 𝐹⟶On)
10 fdm 4993 . . . . . 6 (𝐹:AB → dom 𝐹 = A)
1110eqcomd 2042 . . . . 5 (𝐹:ABA = dom 𝐹)
12 ordeq 4075 . . . . 5 (A = dom 𝐹 → (Ord A ↔ Ord dom 𝐹))
131, 2, 11, 124syl 18 . . . 4 (𝐹 Isom E , E (A, B) → (Ord A ↔ Ord dom 𝐹))
1413biimpa 280 . . 3 ((𝐹 Isom E , E (A, B) Ord A) → Ord dom 𝐹)
15143adant3 923 . 2 ((𝐹 Isom E , E (A, B) Ord A B ⊆ On) → Ord dom 𝐹)
1610eleq2d 2104 . . . . . . 7 (𝐹:AB → (x dom 𝐹x A))
1710eleq2d 2104 . . . . . . 7 (𝐹:AB → (y dom 𝐹y A))
1816, 17anbi12d 442 . . . . . 6 (𝐹:AB → ((x dom 𝐹 y dom 𝐹) ↔ (x A y A)))
191, 2, 183syl 17 . . . . 5 (𝐹 Isom E , E (A, B) → ((x dom 𝐹 y dom 𝐹) ↔ (x A y A)))
20 epel 4020 . . . . . . . . 9 (x E yx y)
21 isorel 5391 . . . . . . . . 9 ((𝐹 Isom E , E (A, B) (x A y A)) → (x E y ↔ (𝐹x) E (𝐹y)))
2220, 21syl5bbr 183 . . . . . . . 8 ((𝐹 Isom E , E (A, B) (x A y A)) → (x y ↔ (𝐹x) E (𝐹y)))
23 ffn 4989 . . . . . . . . . . 11 (𝐹:AB𝐹 Fn A)
243, 23syl 14 . . . . . . . . . 10 (𝐹 Isom E , E (A, B) → 𝐹 Fn A)
2524adantr 261 . . . . . . . . 9 ((𝐹 Isom E , E (A, B) (x A y A)) → 𝐹 Fn A)
26 simprr 484 . . . . . . . . 9 ((𝐹 Isom E , E (A, B) (x A y A)) → y A)
27 funfvex 5135 . . . . . . . . . . 11 ((Fun 𝐹 y dom 𝐹) → (𝐹y) V)
2827funfni 4942 . . . . . . . . . 10 ((𝐹 Fn A y A) → (𝐹y) V)
29 epelg 4018 . . . . . . . . . 10 ((𝐹y) V → ((𝐹x) E (𝐹y) ↔ (𝐹x) (𝐹y)))
3028, 29syl 14 . . . . . . . . 9 ((𝐹 Fn A y A) → ((𝐹x) E (𝐹y) ↔ (𝐹x) (𝐹y)))
3125, 26, 30syl2anc 391 . . . . . . . 8 ((𝐹 Isom E , E (A, B) (x A y A)) → ((𝐹x) E (𝐹y) ↔ (𝐹x) (𝐹y)))
3222, 31bitrd 177 . . . . . . 7 ((𝐹 Isom E , E (A, B) (x A y A)) → (x y ↔ (𝐹x) (𝐹y)))
3332biimpd 132 . . . . . 6 ((𝐹 Isom E , E (A, B) (x A y A)) → (x y → (𝐹x) (𝐹y)))
3433ex 108 . . . . 5 (𝐹 Isom E , E (A, B) → ((x A y A) → (x y → (𝐹x) (𝐹y))))
3519, 34sylbid 139 . . . 4 (𝐹 Isom E , E (A, B) → ((x dom 𝐹 y dom 𝐹) → (x y → (𝐹x) (𝐹y))))
3635ralrimivv 2394 . . 3 (𝐹 Isom E , E (A, B) → x dom 𝐹y dom 𝐹(x y → (𝐹x) (𝐹y)))
37363ad2ant1 924 . 2 ((𝐹 Isom E , E (A, B) Ord A B ⊆ On) → x dom 𝐹y dom 𝐹(x y → (𝐹x) (𝐹y)))
38 df-smo 5842 . 2 (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On Ord dom 𝐹 x dom 𝐹y dom 𝐹(x y → (𝐹x) (𝐹y))))
399, 15, 37, 38syl3anbrc 1087 1 ((𝐹 Isom E , E (A, B) Ord A B ⊆ On) → Smo 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884   = wceq 1242   wcel 1390  wral 2300  Vcvv 2551  wss 2911   class class class wbr 3755   E cep 4015  Ord word 4065  Oncon0 4066  dom cdm 4288   Fn wfn 4840  wf 4841  1-1-ontowf1o 4844  cfv 4845   Isom wiso 4846  Smo wsmo 5841
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-tr 3846  df-eprel 4017  df-id 4021  df-iord 4069  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-f1o 4852  df-fv 4853  df-isom 4854  df-smo 5842
This theorem is referenced by: (None)
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