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Theorem isoti 6943
 Description: An isomorphism preserves tightness. (Contributed by Jim Kingdon, 26-Nov-2021.)
Assertion
Ref Expression
isoti (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)) ↔ ∀𝑢𝐵𝑣𝐵 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢))))
Distinct variable groups:   𝑢,𝐴,𝑣   𝑢,𝐵,𝑣   𝑢,𝐹,𝑣   𝑢,𝑅,𝑣   𝑢,𝑆,𝑣

Proof of Theorem isoti
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isocnv 5756 . . . 4 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹 Isom 𝑆, 𝑅 (𝐵, 𝐴))
2 isotilem 6942 . . . 4 (𝐹 Isom 𝑆, 𝑅 (𝐵, 𝐴) → (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)) → ∀𝑥𝐵𝑦𝐵 (𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥))))
31, 2syl 14 . . 3 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)) → ∀𝑥𝐵𝑦𝐵 (𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥))))
4 isotilem 6942 . . 3 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑥𝐵𝑦𝐵 (𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥)) → ∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))))
53, 4impbid 128 . 2 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥))))
6 equequ1 1692 . . . 4 (𝑥 = 𝑢 → (𝑥 = 𝑦𝑢 = 𝑦))
7 breq1 3968 . . . . . 6 (𝑥 = 𝑢 → (𝑥𝑆𝑦𝑢𝑆𝑦))
87notbid 657 . . . . 5 (𝑥 = 𝑢 → (¬ 𝑥𝑆𝑦 ↔ ¬ 𝑢𝑆𝑦))
9 breq2 3969 . . . . . 6 (𝑥 = 𝑢 → (𝑦𝑆𝑥𝑦𝑆𝑢))
109notbid 657 . . . . 5 (𝑥 = 𝑢 → (¬ 𝑦𝑆𝑥 ↔ ¬ 𝑦𝑆𝑢))
118, 10anbi12d 465 . . . 4 (𝑥 = 𝑢 → ((¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥) ↔ (¬ 𝑢𝑆𝑦 ∧ ¬ 𝑦𝑆𝑢)))
126, 11bibi12d 234 . . 3 (𝑥 = 𝑢 → ((𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥)) ↔ (𝑢 = 𝑦 ↔ (¬ 𝑢𝑆𝑦 ∧ ¬ 𝑦𝑆𝑢))))
13 equequ2 1693 . . . 4 (𝑦 = 𝑣 → (𝑢 = 𝑦𝑢 = 𝑣))
14 breq2 3969 . . . . . 6 (𝑦 = 𝑣 → (𝑢𝑆𝑦𝑢𝑆𝑣))
1514notbid 657 . . . . 5 (𝑦 = 𝑣 → (¬ 𝑢𝑆𝑦 ↔ ¬ 𝑢𝑆𝑣))
16 breq1 3968 . . . . . 6 (𝑦 = 𝑣 → (𝑦𝑆𝑢𝑣𝑆𝑢))
1716notbid 657 . . . . 5 (𝑦 = 𝑣 → (¬ 𝑦𝑆𝑢 ↔ ¬ 𝑣𝑆𝑢))
1815, 17anbi12d 465 . . . 4 (𝑦 = 𝑣 → ((¬ 𝑢𝑆𝑦 ∧ ¬ 𝑦𝑆𝑢) ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢)))
1913, 18bibi12d 234 . . 3 (𝑦 = 𝑣 → ((𝑢 = 𝑦 ↔ (¬ 𝑢𝑆𝑦 ∧ ¬ 𝑦𝑆𝑢)) ↔ (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢))))
2012, 19cbvral2v 2691 . 2 (∀𝑥𝐵𝑦𝐵 (𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥)) ↔ ∀𝑢𝐵𝑣𝐵 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢)))
215, 20bitrdi 195 1 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)) ↔ ∀𝑢𝐵𝑣𝐵 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢))))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104  ∀wral 2435   class class class wbr 3965  ◡ccnv 4582   Isom wiso 5168 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4252  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-f1 5172  df-fo 5173  df-f1o 5174  df-fv 5175  df-isom 5176 This theorem is referenced by:  supisoti  6946
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