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Theorem isores2 7369
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 6862 . . . . . . . 8 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴𝐵)
2 ffvelcdm 7115 . . . . . . . . . 10 ((𝐻:𝐴𝐵𝑥𝐴) → (𝐻𝑥) ∈ 𝐵)
32adantrr 716 . . . . . . . . 9 ((𝐻:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (𝐻𝑥) ∈ 𝐵)
4 ffvelcdm 7115 . . . . . . . . . 10 ((𝐻:𝐴𝐵𝑦𝐴) → (𝐻𝑦) ∈ 𝐵)
54adantrl 715 . . . . . . . . 9 ((𝐻:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (𝐻𝑦) ∈ 𝐵)
6 brinxp 5778 . . . . . . . . 9 (((𝐻𝑥) ∈ 𝐵 ∧ (𝐻𝑦) ∈ 𝐵) → ((𝐻𝑥)𝑆(𝐻𝑦) ↔ (𝐻𝑥)(𝑆 ∩ (𝐵 × 𝐵))(𝐻𝑦)))
73, 5, 6syl2anc 583 . . . . . . . 8 ((𝐻:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → ((𝐻𝑥)𝑆(𝐻𝑦) ↔ (𝐻𝑥)(𝑆 ∩ (𝐵 × 𝐵))(𝐻𝑦)))
81, 7sylan 579 . . . . . . 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 3182 . . . 4 ((𝐻:𝐴1-1-onto𝐵𝑥𝐴) → (∀𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ ∀𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)(𝑆 ∩ (𝐵 × 𝐵))(𝐻𝑦))))
1211ralbidva 3182 . . 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 6582 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
15 df-isom 6582 . 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 2108  wral 3067  cin 3975   class class class wbr 5166   × cxp 5698  wf 6569  1-1-ontowf1o 6572  cfv 6573   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-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-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-f1o 6580  df-fv 6581  df-isom 6582
This theorem is referenced by:  isores1  7370  hartogslem1  9611  leiso  14508  icopnfhmeo  24993  iccpnfhmeo  24995  gtiso  32712  xrge0iifhmeo  33882
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