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Theorem smoiso2 8409
Description: The strictly monotone ordinal functions are also isomorphisms of subclasses of On equipped with the membership relation. (Contributed by Mario Carneiro, 20-Mar-2013.)
Assertion
Ref Expression
smoiso2 ((Ord 𝐴𝐵 ⊆ On) → ((𝐹:𝐴onto𝐵 ∧ Smo 𝐹) ↔ 𝐹 Isom E , E (𝐴, 𝐵)))

Proof of Theorem smoiso2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fof 6820 . . . . . . 7 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
2 smo11 8404 . . . . . . 7 ((𝐹:𝐴𝐵 ∧ Smo 𝐹) → 𝐹:𝐴1-1𝐵)
31, 2sylan 580 . . . . . 6 ((𝐹:𝐴onto𝐵 ∧ Smo 𝐹) → 𝐹:𝐴1-1𝐵)
4 simpl 482 . . . . . 6 ((𝐹:𝐴onto𝐵 ∧ Smo 𝐹) → 𝐹:𝐴onto𝐵)
5 df-f1o 6568 . . . . . 6 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵))
63, 4, 5sylanbrc 583 . . . . 5 ((𝐹:𝐴onto𝐵 ∧ Smo 𝐹) → 𝐹:𝐴1-1-onto𝐵)
76adantl 481 . . . 4 (((Ord 𝐴𝐵 ⊆ On) ∧ (𝐹:𝐴onto𝐵 ∧ Smo 𝐹)) → 𝐹:𝐴1-1-onto𝐵)
8 fofn 6822 . . . . . 6 (𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
9 smoord 8405 . . . . . . . 8 (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑦 ↔ (𝐹𝑥) ∈ (𝐹𝑦)))
10 epel 5587 . . . . . . . 8 (𝑥 E 𝑦𝑥𝑦)
11 fvex 6919 . . . . . . . . 9 (𝐹𝑦) ∈ V
1211epeli 5586 . . . . . . . 8 ((𝐹𝑥) E (𝐹𝑦) ↔ (𝐹𝑥) ∈ (𝐹𝑦))
139, 10, 123bitr4g 314 . . . . . . 7 (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥 E 𝑦 ↔ (𝐹𝑥) E (𝐹𝑦)))
1413ralrimivva 3202 . . . . . 6 ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → ∀𝑥𝐴𝑦𝐴 (𝑥 E 𝑦 ↔ (𝐹𝑥) E (𝐹𝑦)))
158, 14sylan 580 . . . . 5 ((𝐹:𝐴onto𝐵 ∧ Smo 𝐹) → ∀𝑥𝐴𝑦𝐴 (𝑥 E 𝑦 ↔ (𝐹𝑥) E (𝐹𝑦)))
1615adantl 481 . . . 4 (((Ord 𝐴𝐵 ⊆ On) ∧ (𝐹:𝐴onto𝐵 ∧ Smo 𝐹)) → ∀𝑥𝐴𝑦𝐴 (𝑥 E 𝑦 ↔ (𝐹𝑥) E (𝐹𝑦)))
17 df-isom 6570 . . . 4 (𝐹 Isom E , E (𝐴, 𝐵) ↔ (𝐹:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 E 𝑦 ↔ (𝐹𝑥) E (𝐹𝑦))))
187, 16, 17sylanbrc 583 . . 3 (((Ord 𝐴𝐵 ⊆ On) ∧ (𝐹:𝐴onto𝐵 ∧ Smo 𝐹)) → 𝐹 Isom E , E (𝐴, 𝐵))
1918ex 412 . 2 ((Ord 𝐴𝐵 ⊆ On) → ((𝐹:𝐴onto𝐵 ∧ Smo 𝐹) → 𝐹 Isom E , E (𝐴, 𝐵)))
20 isof1o 7343 . . . . . . 7 (𝐹 Isom E , E (𝐴, 𝐵) → 𝐹:𝐴1-1-onto𝐵)
21 f1ofo 6855 . . . . . . 7 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴onto𝐵)
2220, 21syl 17 . . . . . 6 (𝐹 Isom E , E (𝐴, 𝐵) → 𝐹:𝐴onto𝐵)
23223ad2ant1 1134 . . . . 5 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴𝐵 ⊆ On) → 𝐹:𝐴onto𝐵)
24 smoiso 8402 . . . . 5 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴𝐵 ⊆ On) → Smo 𝐹)
2523, 24jca 511 . . . 4 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴𝐵 ⊆ On) → (𝐹:𝐴onto𝐵 ∧ Smo 𝐹))
26253expib 1123 . . 3 (𝐹 Isom E , E (𝐴, 𝐵) → ((Ord 𝐴𝐵 ⊆ On) → (𝐹:𝐴onto𝐵 ∧ Smo 𝐹)))
2726com12 32 . 2 ((Ord 𝐴𝐵 ⊆ On) → (𝐹 Isom E , E (𝐴, 𝐵) → (𝐹:𝐴onto𝐵 ∧ Smo 𝐹)))
2819, 27impbid 212 1 ((Ord 𝐴𝐵 ⊆ On) → ((𝐹:𝐴onto𝐵 ∧ Smo 𝐹) ↔ 𝐹 Isom E , E (𝐴, 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087  wcel 2108  wral 3061  wss 3951   class class class wbr 5143   E cep 5583  Ord word 6383  Oncon0 6384   Fn wfn 6556  wf 6557  1-1wf1 6558  ontowfo 6559  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-10 2141  ax-11 2157  ax-12 2177  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-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-ord 6387  df-on 6388  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-smo 8386
This theorem is referenced by:  oismo  9580  cofsmo  10309
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