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Theorem isores2 7353
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
isores2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, (𝑆 ∩ (𝐵 × 𝐵))(𝐴, 𝐵))

Proof of Theorem isores2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1of 6849 . . . . . . . 8 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴𝐵)
2 ffvelcdm 7101 . . . . . . . . . 10 ((𝐻:𝐴𝐵𝑥𝐴) → (𝐻𝑥) ∈ 𝐵)
32adantrr 717 . . . . . . . . 9 ((𝐻:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (𝐻𝑥) ∈ 𝐵)
4 ffvelcdm 7101 . . . . . . . . . 10 ((𝐻:𝐴𝐵𝑦𝐴) → (𝐻𝑦) ∈ 𝐵)
54adantrl 716 . . . . . . . . 9 ((𝐻:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (𝐻𝑦) ∈ 𝐵)
6 brinxp 5767 . . . . . . . . 9 (((𝐻𝑥) ∈ 𝐵 ∧ (𝐻𝑦) ∈ 𝐵) → ((𝐻𝑥)𝑆(𝐻𝑦) ↔ (𝐻𝑥)(𝑆 ∩ (𝐵 × 𝐵))(𝐻𝑦)))
73, 5, 6syl2anc 584 . . . . . . . 8 ((𝐻:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → ((𝐻𝑥)𝑆(𝐻𝑦) ↔ (𝐻𝑥)(𝑆 ∩ (𝐵 × 𝐵))(𝐻𝑦)))
81, 7sylan 580 . . . . . . 7 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐴𝑦𝐴)) → ((𝐻𝑥)𝑆(𝐻𝑦) ↔ (𝐻𝑥)(𝑆 ∩ (𝐵 × 𝐵))(𝐻𝑦)))
98anassrs 467 . . . . . 6 (((𝐻:𝐴1-1-onto𝐵𝑥𝐴) ∧ 𝑦𝐴) → ((𝐻𝑥)𝑆(𝐻𝑦) ↔ (𝐻𝑥)(𝑆 ∩ (𝐵 × 𝐵))(𝐻𝑦)))
109bibi2d 342 . . . . 5 (((𝐻:𝐴1-1-onto𝐵𝑥𝐴) ∧ 𝑦𝐴) → ((𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ (𝑥𝑅𝑦 ↔ (𝐻𝑥)(𝑆 ∩ (𝐵 × 𝐵))(𝐻𝑦))))
1110ralbidva 3174 . . . 4 ((𝐻:𝐴1-1-onto𝐵𝑥𝐴) → (∀𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ ∀𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)(𝑆 ∩ (𝐵 × 𝐵))(𝐻𝑦))))
1211ralbidva 3174 . . 3 (𝐻:𝐴1-1-onto𝐵 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)(𝑆 ∩ (𝐵 × 𝐵))(𝐻𝑦))))
1312pm5.32i 574 . 2 ((𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)(𝑆 ∩ (𝐵 × 𝐵))(𝐻𝑦))))
14 df-isom 6572 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
15 df-isom 6572 . 2 (𝐻 Isom 𝑅, (𝑆 ∩ (𝐵 × 𝐵))(𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)(𝑆 ∩ (𝐵 × 𝐵))(𝐻𝑦))))
1613, 14, 153bitr4i 303 1 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, (𝑆 ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wcel 2106  wral 3059  cin 3962   class class class wbr 5148   × cxp 5687  wf 6559  1-1-ontowf1o 6562  cfv 6563   Isom wiso 6564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-f1o 6570  df-fv 6571  df-isom 6572
This theorem is referenced by:  isores1  7354  hartogslem1  9580  leiso  14495  icopnfhmeo  24988  iccpnfhmeo  24990  gtiso  32716  xrge0iifhmeo  33897
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