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Theorem isores1 7370
Description: An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)
Assertion
Ref Expression
isores1 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵))

Proof of Theorem isores1
StepHypRef Expression
1 isocnv 7366 . . . . 5 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴))
2 isores2 7369 . . . . 5 (𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴) ↔ 𝐻 Isom 𝑆, (𝑅 ∩ (𝐴 × 𝐴))(𝐵, 𝐴))
31, 2sylib 218 . . . 4 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻 Isom 𝑆, (𝑅 ∩ (𝐴 × 𝐴))(𝐵, 𝐴))
4 isocnv 7366 . . . 4 (𝐻 Isom 𝑆, (𝑅 ∩ (𝐴 × 𝐴))(𝐵, 𝐴) → 𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵))
53, 4syl 17 . . 3 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵))
6 isof1o 7359 . . . 4 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
7 f1orel 6865 . . . 4 (𝐻:𝐴1-1-onto𝐵 → Rel 𝐻)
8 dfrel2 6220 . . . . 5 (Rel 𝐻𝐻 = 𝐻)
9 isoeq1 7353 . . . . 5 (𝐻 = 𝐻 → (𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵) ↔ 𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵)))
108, 9sylbi 217 . . . 4 (Rel 𝐻 → (𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵) ↔ 𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵)))
116, 7, 103syl 18 . . 3 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵) ↔ 𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵)))
125, 11mpbid 232 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵))
13 isocnv 7366 . . . . 5 (𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵) → 𝐻 Isom 𝑆, (𝑅 ∩ (𝐴 × 𝐴))(𝐵, 𝐴))
1413, 2sylibr 234 . . . 4 (𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵) → 𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴))
15 isocnv 7366 . . . 4 (𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
1614, 15syl 17 . . 3 (𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
17 isof1o 7359 . . . 4 (𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
18 isoeq1 7353 . . . . 5 (𝐻 = 𝐻 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)))
198, 18sylbi 217 . . . 4 (Rel 𝐻 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)))
2017, 7, 193syl 18 . . 3 (𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵) → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)))
2116, 20mpbid 232 . 2 (𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
2212, 21impbii 209 1 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1537  cin 3975   × cxp 5698  ccnv 5699  Rel wrel 5705  1-1-ontowf1o 6572   Isom wiso 6574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582
This theorem is referenced by:  leiso  14508  icopnfhmeo  24993  iccpnfhmeo  24995  xrhmeo  24996  gtiso  32712  xrge0iifhmeo  33882
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