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Theorem smoiso 6081
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 𝐴𝐵 ⊆ On) → Smo 𝐹)

Proof of Theorem smoiso
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isof1o 5600 . . . 4 (𝐹 Isom E , E (𝐴, 𝐵) → 𝐹:𝐴1-1-onto𝐵)
2 f1of 5266 . . . 4 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)
31, 2syl 14 . . 3 (𝐹 Isom E , E (𝐴, 𝐵) → 𝐹:𝐴𝐵)
4 ffdm 5194 . . . . . 6 (𝐹:𝐴𝐵 → (𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴))
54simpld 111 . . . . 5 (𝐹:𝐴𝐵𝐹:dom 𝐹𝐵)
6 fss 5185 . . . . 5 ((𝐹:dom 𝐹𝐵𝐵 ⊆ On) → 𝐹:dom 𝐹⟶On)
75, 6sylan 278 . . . 4 ((𝐹:𝐴𝐵𝐵 ⊆ On) → 𝐹:dom 𝐹⟶On)
873adant2 963 . . 3 ((𝐹:𝐴𝐵 ∧ Ord 𝐴𝐵 ⊆ On) → 𝐹:dom 𝐹⟶On)
93, 8syl3an1 1208 . 2 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴𝐵 ⊆ On) → 𝐹:dom 𝐹⟶On)
10 fdm 5179 . . . . . 6 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
1110eqcomd 2094 . . . . 5 (𝐹:𝐴𝐵𝐴 = dom 𝐹)
12 ordeq 4208 . . . . 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 291 . . 3 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴) → Ord dom 𝐹)
15143adant3 964 . 2 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴𝐵 ⊆ On) → Ord dom 𝐹)
1610eleq2d 2158 . . . . . . 7 (𝐹:𝐴𝐵 → (𝑥 ∈ dom 𝐹𝑥𝐴))
1710eleq2d 2158 . . . . . . 7 (𝐹:𝐴𝐵 → (𝑦 ∈ dom 𝐹𝑦𝐴))
1816, 17anbi12d 458 . . . . . 6 (𝐹:𝐴𝐵 → ((𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹) ↔ (𝑥𝐴𝑦𝐴)))
191, 2, 183syl 17 . . . . 5 (𝐹 Isom E , E (𝐴, 𝐵) → ((𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹) ↔ (𝑥𝐴𝑦𝐴)))
20 epel 4128 . . . . . . . . 9 (𝑥 E 𝑦𝑥𝑦)
21 isorel 5601 . . . . . . . . 9 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥 E 𝑦 ↔ (𝐹𝑥) E (𝐹𝑦)))
2220, 21syl5bbr 193 . . . . . . . 8 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑦 ↔ (𝐹𝑥) E (𝐹𝑦)))
23 ffn 5174 . . . . . . . . . . 11 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
243, 23syl 14 . . . . . . . . . 10 (𝐹 Isom E , E (𝐴, 𝐵) → 𝐹 Fn 𝐴)
2524adantr 271 . . . . . . . . 9 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ (𝑥𝐴𝑦𝐴)) → 𝐹 Fn 𝐴)
26 simprr 500 . . . . . . . . 9 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ (𝑥𝐴𝑦𝐴)) → 𝑦𝐴)
27 funfvex 5335 . . . . . . . . . . 11 ((Fun 𝐹𝑦 ∈ dom 𝐹) → (𝐹𝑦) ∈ V)
2827funfni 5127 . . . . . . . . . 10 ((𝐹 Fn 𝐴𝑦𝐴) → (𝐹𝑦) ∈ V)
29 epelg 4126 . . . . . . . . . 10 ((𝐹𝑦) ∈ V → ((𝐹𝑥) E (𝐹𝑦) ↔ (𝐹𝑥) ∈ (𝐹𝑦)))
3028, 29syl 14 . . . . . . . . 9 ((𝐹 Fn 𝐴𝑦𝐴) → ((𝐹𝑥) E (𝐹𝑦) ↔ (𝐹𝑥) ∈ (𝐹𝑦)))
3125, 26, 30syl2anc 404 . . . . . . . 8 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ (𝑥𝐴𝑦𝐴)) → ((𝐹𝑥) E (𝐹𝑦) ↔ (𝐹𝑥) ∈ (𝐹𝑦)))
3222, 31bitrd 187 . . . . . . 7 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑦 ↔ (𝐹𝑥) ∈ (𝐹𝑦)))
3332biimpd 143 . . . . . 6 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑦 → (𝐹𝑥) ∈ (𝐹𝑦)))
3433ex 114 . . . . 5 (𝐹 Isom E , E (𝐴, 𝐵) → ((𝑥𝐴𝑦𝐴) → (𝑥𝑦 → (𝐹𝑥) ∈ (𝐹𝑦))))
3519, 34sylbid 149 . . . 4 (𝐹 Isom E , E (𝐴, 𝐵) → ((𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹) → (𝑥𝑦 → (𝐹𝑥) ∈ (𝐹𝑦))))
3635ralrimivv 2455 . . 3 (𝐹 Isom E , E (𝐴, 𝐵) → ∀𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹(𝑥𝑦 → (𝐹𝑥) ∈ (𝐹𝑦)))
37363ad2ant1 965 . 2 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴𝐵 ⊆ On) → ∀𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹(𝑥𝑦 → (𝐹𝑥) ∈ (𝐹𝑦)))
38 df-smo 6065 . 2 (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹(𝑥𝑦 → (𝐹𝑥) ∈ (𝐹𝑦))))
399, 15, 37, 38syl3anbrc 1128 1 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴𝐵 ⊆ On) → Smo 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 925   = wceq 1290  wcel 1439  wral 2360  Vcvv 2620  wss 3000   class class class wbr 3851   E cep 4123  Ord word 4198  Oncon0 4199  dom cdm 4452   Fn wfn 5023  wf 5024  1-1-ontowf1o 5027  cfv 5028   Isom wiso 5029  Smo wsmo 6064
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-tr 3943  df-eprel 4125  df-id 4129  df-iord 4202  df-cnv 4460  df-co 4461  df-dm 4462  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-f1o 5035  df-fv 5036  df-isom 5037  df-smo 6065
This theorem is referenced by: (None)
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