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Theorem smoiso 8402
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 7343 . . . 4 (𝐹 Isom E , E (𝐴, 𝐵) → 𝐹:𝐴1-1-onto𝐵)
2 f1of 6848 . . . 4 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)
31, 2syl 17 . . 3 (𝐹 Isom E , E (𝐴, 𝐵) → 𝐹:𝐴𝐵)
4 ffdm 6765 . . . . . 6 (𝐹:𝐴𝐵 → (𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴))
54simpld 494 . . . . 5 (𝐹:𝐴𝐵𝐹:dom 𝐹𝐵)
6 fss 6752 . . . . 5 ((𝐹:dom 𝐹𝐵𝐵 ⊆ On) → 𝐹:dom 𝐹⟶On)
75, 6sylan 580 . . . 4 ((𝐹:𝐴𝐵𝐵 ⊆ On) → 𝐹:dom 𝐹⟶On)
873adant2 1132 . . 3 ((𝐹:𝐴𝐵 ∧ Ord 𝐴𝐵 ⊆ On) → 𝐹:dom 𝐹⟶On)
93, 8syl3an1 1164 . 2 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴𝐵 ⊆ On) → 𝐹:dom 𝐹⟶On)
10 fdm 6745 . . . . . 6 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
1110eqcomd 2743 . . . . 5 (𝐹:𝐴𝐵𝐴 = dom 𝐹)
12 ordeq 6391 . . . . 5 (𝐴 = dom 𝐹 → (Ord 𝐴 ↔ Ord dom 𝐹))
131, 2, 11, 124syl 19 . . . 4 (𝐹 Isom E , E (𝐴, 𝐵) → (Ord 𝐴 ↔ Ord dom 𝐹))
1413biimpa 476 . . 3 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴) → Ord dom 𝐹)
15143adant3 1133 . 2 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴𝐵 ⊆ On) → Ord dom 𝐹)
1610eleq2d 2827 . . . . . . 7 (𝐹:𝐴𝐵 → (𝑥 ∈ dom 𝐹𝑥𝐴))
1710eleq2d 2827 . . . . . . 7 (𝐹:𝐴𝐵 → (𝑦 ∈ dom 𝐹𝑦𝐴))
1816, 17anbi12d 632 . . . . . 6 (𝐹:𝐴𝐵 → ((𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹) ↔ (𝑥𝐴𝑦𝐴)))
191, 2, 183syl 18 . . . . 5 (𝐹 Isom E , E (𝐴, 𝐵) → ((𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹) ↔ (𝑥𝐴𝑦𝐴)))
20 isorel 7346 . . . . . . . 8 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥 E 𝑦 ↔ (𝐹𝑥) E (𝐹𝑦)))
21 epel 5587 . . . . . . . 8 (𝑥 E 𝑦𝑥𝑦)
22 fvex 6919 . . . . . . . . 9 (𝐹𝑦) ∈ V
2322epeli 5586 . . . . . . . 8 ((𝐹𝑥) E (𝐹𝑦) ↔ (𝐹𝑥) ∈ (𝐹𝑦))
2420, 21, 233bitr3g 313 . . . . . . 7 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑦 ↔ (𝐹𝑥) ∈ (𝐹𝑦)))
2524biimpd 229 . . . . . 6 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑦 → (𝐹𝑥) ∈ (𝐹𝑦)))
2625ex 412 . . . . 5 (𝐹 Isom E , E (𝐴, 𝐵) → ((𝑥𝐴𝑦𝐴) → (𝑥𝑦 → (𝐹𝑥) ∈ (𝐹𝑦))))
2719, 26sylbid 240 . . . 4 (𝐹 Isom E , E (𝐴, 𝐵) → ((𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹) → (𝑥𝑦 → (𝐹𝑥) ∈ (𝐹𝑦))))
2827ralrimivv 3200 . . 3 (𝐹 Isom E , E (𝐴, 𝐵) → ∀𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹(𝑥𝑦 → (𝐹𝑥) ∈ (𝐹𝑦)))
29283ad2ant1 1134 . 2 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴𝐵 ⊆ On) → ∀𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹(𝑥𝑦 → (𝐹𝑥) ∈ (𝐹𝑦)))
30 df-smo 8386 . 2 (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹(𝑥𝑦 → (𝐹𝑥) ∈ (𝐹𝑦))))
319, 15, 29, 30syl3anbrc 1344 1 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴𝐵 ⊆ On) → Smo 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wss 3951   class class class wbr 5143   E cep 5583  dom cdm 5685  Ord word 6383  Oncon0 6384  wf 6557  1-1-ontowf1o 6560  cfv 6561   Isom wiso 6562  Smo wsmo 8385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-iota 6514  df-fn 6564  df-f 6565  df-f1 6566  df-f1o 6568  df-fv 6569  df-isom 6570  df-smo 8386
This theorem is referenced by:  smoiso2  8409
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