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Theorem smoiso2 8035
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 6592 . . . . . . 7 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
2 smo11 8030 . . . . . . 7 ((𝐹:𝐴𝐵 ∧ Smo 𝐹) → 𝐹:𝐴1-1𝐵)
31, 2sylan 583 . . . . . 6 ((𝐹:𝐴onto𝐵 ∧ Smo 𝐹) → 𝐹:𝐴1-1𝐵)
4 simpl 486 . . . . . 6 ((𝐹:𝐴onto𝐵 ∧ Smo 𝐹) → 𝐹:𝐴onto𝐵)
5 df-f1o 6346 . . . . . 6 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵))
63, 4, 5sylanbrc 586 . . . . 5 ((𝐹:𝐴onto𝐵 ∧ Smo 𝐹) → 𝐹:𝐴1-1-onto𝐵)
76adantl 485 . . . 4 (((Ord 𝐴𝐵 ⊆ On) ∧ (𝐹:𝐴onto𝐵 ∧ Smo 𝐹)) → 𝐹:𝐴1-1-onto𝐵)
8 fofn 6594 . . . . . 6 (𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
9 smoord 8031 . . . . . . . 8 (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑦 ↔ (𝐹𝑥) ∈ (𝐹𝑦)))
10 epel 5437 . . . . . . . 8 (𝑥 E 𝑦𝑥𝑦)
11 fvex 6687 . . . . . . . . 9 (𝐹𝑦) ∈ V
1211epeli 5436 . . . . . . . 8 ((𝐹𝑥) E (𝐹𝑦) ↔ (𝐹𝑥) ∈ (𝐹𝑦))
139, 10, 123bitr4g 317 . . . . . . 7 (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥 E 𝑦 ↔ (𝐹𝑥) E (𝐹𝑦)))
1413ralrimivva 3103 . . . . . 6 ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → ∀𝑥𝐴𝑦𝐴 (𝑥 E 𝑦 ↔ (𝐹𝑥) E (𝐹𝑦)))
158, 14sylan 583 . . . . 5 ((𝐹:𝐴onto𝐵 ∧ Smo 𝐹) → ∀𝑥𝐴𝑦𝐴 (𝑥 E 𝑦 ↔ (𝐹𝑥) E (𝐹𝑦)))
1615adantl 485 . . . 4 (((Ord 𝐴𝐵 ⊆ On) ∧ (𝐹:𝐴onto𝐵 ∧ Smo 𝐹)) → ∀𝑥𝐴𝑦𝐴 (𝑥 E 𝑦 ↔ (𝐹𝑥) E (𝐹𝑦)))
17 df-isom 6348 . . . 4 (𝐹 Isom E , E (𝐴, 𝐵) ↔ (𝐹:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 E 𝑦 ↔ (𝐹𝑥) E (𝐹𝑦))))
187, 16, 17sylanbrc 586 . . 3 (((Ord 𝐴𝐵 ⊆ On) ∧ (𝐹:𝐴onto𝐵 ∧ Smo 𝐹)) → 𝐹 Isom E , E (𝐴, 𝐵))
1918ex 416 . 2 ((Ord 𝐴𝐵 ⊆ On) → ((𝐹:𝐴onto𝐵 ∧ Smo 𝐹) → 𝐹 Isom E , E (𝐴, 𝐵)))
20 isof1o 7089 . . . . . . 7 (𝐹 Isom E , E (𝐴, 𝐵) → 𝐹:𝐴1-1-onto𝐵)
21 f1ofo 6625 . . . . . . 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 8028 . . . . 5 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴𝐵 ⊆ On) → Smo 𝐹)
2523, 24jca 515 . . . 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 215 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 209  wa 399  w3a 1088  wcel 2114  wral 3053  wss 3843   class class class wbr 5030   E cep 5433  Ord word 6171  Oncon0 6172   Fn wfn 6334  wf 6335  1-1wf1 6336  ontowfo 6337  1-1-ontowf1o 6338  cfv 6339   Isom wiso 6340  Smo wsmo 8011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-ord 6175  df-on 6176  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-isom 6348  df-smo 8012
This theorem is referenced by:  oismo  9077  cofsmo  9769
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