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Theorem isores2 7347
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 6844 . . . . . . . 8 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴𝐵)
2 ffvelcdm 7096 . . . . . . . . . 10 ((𝐻:𝐴𝐵𝑥𝐴) → (𝐻𝑥) ∈ 𝐵)
32adantrr 715 . . . . . . . . 9 ((𝐻:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (𝐻𝑥) ∈ 𝐵)
4 ffvelcdm 7096 . . . . . . . . . 10 ((𝐻:𝐴𝐵𝑦𝐴) → (𝐻𝑦) ∈ 𝐵)
54adantrl 714 . . . . . . . . 9 ((𝐻:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (𝐻𝑦) ∈ 𝐵)
6 brinxp 5760 . . . . . . . . 9 (((𝐻𝑥) ∈ 𝐵 ∧ (𝐻𝑦) ∈ 𝐵) → ((𝐻𝑥)𝑆(𝐻𝑦) ↔ (𝐻𝑥)(𝑆 ∩ (𝐵 × 𝐵))(𝐻𝑦)))
73, 5, 6syl2anc 582 . . . . . . . 8 ((𝐻:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → ((𝐻𝑥)𝑆(𝐻𝑦) ↔ (𝐻𝑥)(𝑆 ∩ (𝐵 × 𝐵))(𝐻𝑦)))
81, 7sylan 578 . . . . . . 7 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐴𝑦𝐴)) → ((𝐻𝑥)𝑆(𝐻𝑦) ↔ (𝐻𝑥)(𝑆 ∩ (𝐵 × 𝐵))(𝐻𝑦)))
98anassrs 466 . . . . . 6 (((𝐻:𝐴1-1-onto𝐵𝑥𝐴) ∧ 𝑦𝐴) → ((𝐻𝑥)𝑆(𝐻𝑦) ↔ (𝐻𝑥)(𝑆 ∩ (𝐵 × 𝐵))(𝐻𝑦)))
109bibi2d 341 . . . . 5 (((𝐻:𝐴1-1-onto𝐵𝑥𝐴) ∧ 𝑦𝐴) → ((𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ (𝑥𝑅𝑦 ↔ (𝐻𝑥)(𝑆 ∩ (𝐵 × 𝐵))(𝐻𝑦))))
1110ralbidva 3173 . . . 4 ((𝐻:𝐴1-1-onto𝐵𝑥𝐴) → (∀𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ ∀𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)(𝑆 ∩ (𝐵 × 𝐵))(𝐻𝑦))))
1211ralbidva 3173 . . 3 (𝐻:𝐴1-1-onto𝐵 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)(𝑆 ∩ (𝐵 × 𝐵))(𝐻𝑦))))
1312pm5.32i 573 . 2 ((𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)(𝑆 ∩ (𝐵 × 𝐵))(𝐻𝑦))))
14 df-isom 6562 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
15 df-isom 6562 . 2 (𝐻 Isom 𝑅, (𝑆 ∩ (𝐵 × 𝐵))(𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)(𝑆 ∩ (𝐵 × 𝐵))(𝐻𝑦))))
1613, 14, 153bitr4i 302 1 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, (𝑆 ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394  wcel 2098  wral 3058  cin 3948   class class class wbr 5152   × cxp 5680  wf 6549  1-1-ontowf1o 6552  cfv 6553   Isom wiso 6554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-f1o 6560  df-fv 6561  df-isom 6562
This theorem is referenced by:  isores1  7348  hartogslem1  9573  leiso  14460  icopnfhmeo  24888  iccpnfhmeo  24890  gtiso  32501  xrge0iifhmeo  33570
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