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Theorem smoiso 6208
 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 5717 . . . 4 (𝐹 Isom E , E (𝐴, 𝐵) → 𝐹:𝐴1-1-onto𝐵)
2 f1of 5376 . . . 4 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)
31, 2syl 14 . . 3 (𝐹 Isom E , E (𝐴, 𝐵) → 𝐹:𝐴𝐵)
4 ffdm 5302 . . . . . 6 (𝐹:𝐴𝐵 → (𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴))
54simpld 111 . . . . 5 (𝐹:𝐴𝐵𝐹:dom 𝐹𝐵)
6 fss 5293 . . . . 5 ((𝐹:dom 𝐹𝐵𝐵 ⊆ On) → 𝐹:dom 𝐹⟶On)
75, 6sylan 281 . . . 4 ((𝐹:𝐴𝐵𝐵 ⊆ On) → 𝐹:dom 𝐹⟶On)
873adant2 1001 . . 3 ((𝐹:𝐴𝐵 ∧ Ord 𝐴𝐵 ⊆ On) → 𝐹:dom 𝐹⟶On)
93, 8syl3an1 1250 . 2 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴𝐵 ⊆ On) → 𝐹:dom 𝐹⟶On)
10 fdm 5287 . . . . . 6 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
1110eqcomd 2146 . . . . 5 (𝐹:𝐴𝐵𝐴 = dom 𝐹)
12 ordeq 4303 . . . . 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 294 . . 3 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴) → Ord dom 𝐹)
15143adant3 1002 . 2 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴𝐵 ⊆ On) → Ord dom 𝐹)
1610eleq2d 2210 . . . . . . 7 (𝐹:𝐴𝐵 → (𝑥 ∈ dom 𝐹𝑥𝐴))
1710eleq2d 2210 . . . . . . 7 (𝐹:𝐴𝐵 → (𝑦 ∈ dom 𝐹𝑦𝐴))
1816, 17anbi12d 465 . . . . . 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 4223 . . . . . . . . 9 (𝑥 E 𝑦𝑥𝑦)
21 isorel 5718 . . . . . . . . 9 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥 E 𝑦 ↔ (𝐹𝑥) E (𝐹𝑦)))
2220, 21bitr3id 193 . . . . . . . 8 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑦 ↔ (𝐹𝑥) E (𝐹𝑦)))
23 ffn 5281 . . . . . . . . . . 11 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
243, 23syl 14 . . . . . . . . . 10 (𝐹 Isom E , E (𝐴, 𝐵) → 𝐹 Fn 𝐴)
2524adantr 274 . . . . . . . . 9 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ (𝑥𝐴𝑦𝐴)) → 𝐹 Fn 𝐴)
26 simprr 522 . . . . . . . . 9 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ (𝑥𝐴𝑦𝐴)) → 𝑦𝐴)
27 funfvex 5447 . . . . . . . . . . 11 ((Fun 𝐹𝑦 ∈ dom 𝐹) → (𝐹𝑦) ∈ V)
2827funfni 5232 . . . . . . . . . 10 ((𝐹 Fn 𝐴𝑦𝐴) → (𝐹𝑦) ∈ V)
29 epelg 4221 . . . . . . . . . 10 ((𝐹𝑦) ∈ V → ((𝐹𝑥) E (𝐹𝑦) ↔ (𝐹𝑥) ∈ (𝐹𝑦)))
3028, 29syl 14 . . . . . . . . 9 ((𝐹 Fn 𝐴𝑦𝐴) → ((𝐹𝑥) E (𝐹𝑦) ↔ (𝐹𝑥) ∈ (𝐹𝑦)))
3125, 26, 30syl2anc 409 . . . . . . . 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 2517 . . 3 (𝐹 Isom E , E (𝐴, 𝐵) → ∀𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹(𝑥𝑦 → (𝐹𝑥) ∈ (𝐹𝑦)))
37363ad2ant1 1003 . 2 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴𝐵 ⊆ On) → ∀𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹(𝑥𝑦 → (𝐹𝑥) ∈ (𝐹𝑦)))
38 df-smo 6192 . 2 (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹(𝑥𝑦 → (𝐹𝑥) ∈ (𝐹𝑦))))
399, 15, 37, 38syl3anbrc 1166 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 963   = wceq 1332   ∈ wcel 1481  ∀wral 2417  Vcvv 2690   ⊆ wss 3077   class class class wbr 3938   E cep 4218  Ord word 4293  Oncon0 4294  dom cdm 4548   Fn wfn 5127  ⟶wf 5128  –1-1-onto→wf1o 5131  ‘cfv 5132   Isom wiso 5133  Smo wsmo 6191 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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4055  ax-pow 4107  ax-pr 4140 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2692  df-sbc 2915  df-un 3081  df-in 3083  df-ss 3090  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-br 3939  df-opab 3999  df-tr 4036  df-eprel 4220  df-id 4224  df-iord 4297  df-cnv 4556  df-co 4557  df-dm 4558  df-iota 5097  df-fun 5134  df-fn 5135  df-f 5136  df-f1 5137  df-f1o 5139  df-fv 5140  df-isom 5141  df-smo 6192 This theorem is referenced by: (None)
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