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Theorem isores2 7325
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 6826 . . . . . . . 8 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴𝐵)
2 ffvelcdm 7076 . . . . . . . . . 10 ((𝐻:𝐴𝐵𝑥𝐴) → (𝐻𝑥) ∈ 𝐵)
32adantrr 714 . . . . . . . . 9 ((𝐻:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (𝐻𝑥) ∈ 𝐵)
4 ffvelcdm 7076 . . . . . . . . . 10 ((𝐻:𝐴𝐵𝑦𝐴) → (𝐻𝑦) ∈ 𝐵)
54adantrl 713 . . . . . . . . 9 ((𝐻:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (𝐻𝑦) ∈ 𝐵)
6 brinxp 5747 . . . . . . . . 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 3169 . . . 4 ((𝐻:𝐴1-1-onto𝐵𝑥𝐴) → (∀𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ ∀𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)(𝑆 ∩ (𝐵 × 𝐵))(𝐻𝑦))))
1211ralbidva 3169 . . 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 6545 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
15 df-isom 6545 . 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 205  wa 395  wcel 2098  wral 3055  cin 3942   class class class wbr 5141   × cxp 5667  wf 6532  1-1-ontowf1o 6535  cfv 6536   Isom wiso 6537
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-f1o 6543  df-fv 6544  df-isom 6545
This theorem is referenced by:  isores1  7326  hartogslem1  9536  leiso  14423  icopnfhmeo  24818  iccpnfhmeo  24820  gtiso  32426  xrge0iifhmeo  33445
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