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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 6848 . . . . . . . 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 5764 . . . . . . . . 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 3176 . . . 4 ((𝐻:𝐴1-1-onto𝐵𝑥𝐴) → (∀𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ ∀𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)(𝑆 ∩ (𝐵 × 𝐵))(𝐻𝑦))))
1211ralbidva 3176 . . 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 6570 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
15 df-isom 6570 . 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 3061  cin 3950   class class class wbr 5143   × cxp 5683  wf 6557  1-1-ontowf1o 6560  cfv 6561   Isom wiso 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-f1o 6568  df-fv 6569  df-isom 6570
This theorem is referenced by:  isores1  7354  hartogslem1  9582  leiso  14498  icopnfhmeo  24974  iccpnfhmeo  24976  gtiso  32710  xrge0iifhmeo  33935
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