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Theorem smoiso2 8200
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 6688 . . . . . . 7 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
2 smo11 8195 . . . . . . 7 ((𝐹:𝐴𝐵 ∧ Smo 𝐹) → 𝐹:𝐴1-1𝐵)
31, 2sylan 580 . . . . . 6 ((𝐹:𝐴onto𝐵 ∧ Smo 𝐹) → 𝐹:𝐴1-1𝐵)
4 simpl 483 . . . . . 6 ((𝐹:𝐴onto𝐵 ∧ Smo 𝐹) → 𝐹:𝐴onto𝐵)
5 df-f1o 6440 . . . . . 6 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵))
63, 4, 5sylanbrc 583 . . . . 5 ((𝐹:𝐴onto𝐵 ∧ Smo 𝐹) → 𝐹:𝐴1-1-onto𝐵)
76adantl 482 . . . 4 (((Ord 𝐴𝐵 ⊆ On) ∧ (𝐹:𝐴onto𝐵 ∧ Smo 𝐹)) → 𝐹:𝐴1-1-onto𝐵)
8 fofn 6690 . . . . . 6 (𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
9 smoord 8196 . . . . . . . 8 (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑦 ↔ (𝐹𝑥) ∈ (𝐹𝑦)))
10 epel 5498 . . . . . . . 8 (𝑥 E 𝑦𝑥𝑦)
11 fvex 6787 . . . . . . . . 9 (𝐹𝑦) ∈ V
1211epeli 5497 . . . . . . . 8 ((𝐹𝑥) E (𝐹𝑦) ↔ (𝐹𝑥) ∈ (𝐹𝑦))
139, 10, 123bitr4g 314 . . . . . . 7 (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥 E 𝑦 ↔ (𝐹𝑥) E (𝐹𝑦)))
1413ralrimivva 3123 . . . . . 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 482 . . . 4 (((Ord 𝐴𝐵 ⊆ On) ∧ (𝐹:𝐴onto𝐵 ∧ Smo 𝐹)) → ∀𝑥𝐴𝑦𝐴 (𝑥 E 𝑦 ↔ (𝐹𝑥) E (𝐹𝑦)))
17 df-isom 6442 . . . 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 413 . 2 ((Ord 𝐴𝐵 ⊆ On) → ((𝐹:𝐴onto𝐵 ∧ Smo 𝐹) → 𝐹 Isom E , E (𝐴, 𝐵)))
20 isof1o 7194 . . . . . . 7 (𝐹 Isom E , E (𝐴, 𝐵) → 𝐹:𝐴1-1-onto𝐵)
21 f1ofo 6723 . . . . . . 7 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴onto𝐵)
2220, 21syl 17 . . . . . 6 (𝐹 Isom E , E (𝐴, 𝐵) → 𝐹:𝐴onto𝐵)
23223ad2ant1 1132 . . . . 5 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴𝐵 ⊆ On) → 𝐹:𝐴onto𝐵)
24 smoiso 8193 . . . . 5 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴𝐵 ⊆ On) → Smo 𝐹)
2523, 24jca 512 . . . 4 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴𝐵 ⊆ On) → (𝐹:𝐴onto𝐵 ∧ Smo 𝐹))
26253expib 1121 . . 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 211 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 205  wa 396  w3a 1086  wcel 2106  wral 3064  wss 3887   class class class wbr 5074   E cep 5494  Ord word 6265  Oncon0 6266   Fn wfn 6428  wf 6429  1-1wf1 6430  ontowfo 6431  1-1-ontowf1o 6432  cfv 6433   Isom wiso 6434  Smo wsmo 8176
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-ord 6269  df-on 6270  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-smo 8177
This theorem is referenced by:  oismo  9299  cofsmo  10025
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