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Theorem isoti 6860
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 5678 . . . 4 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹 Isom 𝑆, 𝑅 (𝐵, 𝐴))
2 isotilem 6859 . . . 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 6859 . . 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 1671 . . . 4 (𝑥 = 𝑢 → (𝑥 = 𝑦𝑢 = 𝑦))
7 breq1 3900 . . . . . 6 (𝑥 = 𝑢 → (𝑥𝑆𝑦𝑢𝑆𝑦))
87notbid 639 . . . . 5 (𝑥 = 𝑢 → (¬ 𝑥𝑆𝑦 ↔ ¬ 𝑢𝑆𝑦))
9 breq2 3901 . . . . . 6 (𝑥 = 𝑢 → (𝑦𝑆𝑥𝑦𝑆𝑢))
109notbid 639 . . . . 5 (𝑥 = 𝑢 → (¬ 𝑦𝑆𝑥 ↔ ¬ 𝑦𝑆𝑢))
118, 10anbi12d 462 . . . 4 (𝑥 = 𝑢 → ((¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥) ↔ (¬ 𝑢𝑆𝑦 ∧ ¬ 𝑦𝑆𝑢)))
126, 11bibi12d 234 . . 3 (𝑥 = 𝑢 → ((𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥)) ↔ (𝑢 = 𝑦 ↔ (¬ 𝑢𝑆𝑦 ∧ ¬ 𝑦𝑆𝑢))))
13 equequ2 1672 . . . 4 (𝑦 = 𝑣 → (𝑢 = 𝑦𝑢 = 𝑣))
14 breq2 3901 . . . . . 6 (𝑦 = 𝑣 → (𝑢𝑆𝑦𝑢𝑆𝑣))
1514notbid 639 . . . . 5 (𝑦 = 𝑣 → (¬ 𝑢𝑆𝑦 ↔ ¬ 𝑢𝑆𝑣))
16 breq1 3900 . . . . . 6 (𝑦 = 𝑣 → (𝑦𝑆𝑢𝑣𝑆𝑢))
1716notbid 639 . . . . 5 (𝑦 = 𝑣 → (¬ 𝑦𝑆𝑢 ↔ ¬ 𝑣𝑆𝑢))
1815, 17anbi12d 462 . . . 4 (𝑦 = 𝑣 → ((¬ 𝑢𝑆𝑦 ∧ ¬ 𝑦𝑆𝑢) ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢)))
1913, 18bibi12d 234 . . 3 (𝑦 = 𝑣 → ((𝑢 = 𝑦 ↔ (¬ 𝑢𝑆𝑦 ∧ ¬ 𝑦𝑆𝑢)) ↔ (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢))))
2012, 19cbvral2v 2637 . 2 (∀𝑥𝐵𝑦𝐵 (𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥)) ↔ ∀𝑢𝐵𝑣𝐵 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢)))
215, 20syl6bb 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 2391   class class class wbr 3897  ccnv 4506   Isom wiso 5092
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-isom 5100
This theorem is referenced by:  supisoti  6863
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