ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  isotilem GIF version

Theorem isotilem 6410
Description: Lemma for isoti 6411. (Contributed by Jim Kingdon, 26-Nov-2021.)
Assertion
Ref Expression
isotilem (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑥𝐵𝑦𝐵 (𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥)) → ∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))))
Distinct variable groups:   𝑢,𝐴,𝑣   𝑢,𝐵,𝑣,𝑥,𝑦   𝑢,𝐹,𝑣,𝑥,𝑦   𝑢,𝑅,𝑣   𝑢,𝑆,𝑣,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝑅(𝑥,𝑦)

Proof of Theorem isotilem
StepHypRef Expression
1 isof1o 5475 . . . . . 6 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴1-1-onto𝐵)
2 f1of 5154 . . . . . 6 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)
3 ffvelrn 5328 . . . . . . . 8 ((𝐹:𝐴𝐵𝑢𝐴) → (𝐹𝑢) ∈ 𝐵)
43ex 112 . . . . . . 7 (𝐹:𝐴𝐵 → (𝑢𝐴 → (𝐹𝑢) ∈ 𝐵))
5 ffvelrn 5328 . . . . . . . 8 ((𝐹:𝐴𝐵𝑣𝐴) → (𝐹𝑣) ∈ 𝐵)
65ex 112 . . . . . . 7 (𝐹:𝐴𝐵 → (𝑣𝐴 → (𝐹𝑣) ∈ 𝐵))
74, 6anim12d 322 . . . . . 6 (𝐹:𝐴𝐵 → ((𝑢𝐴𝑣𝐴) → ((𝐹𝑢) ∈ 𝐵 ∧ (𝐹𝑣) ∈ 𝐵)))
81, 2, 73syl 17 . . . . 5 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ((𝑢𝐴𝑣𝐴) → ((𝐹𝑢) ∈ 𝐵 ∧ (𝐹𝑣) ∈ 𝐵)))
98imp 119 . . . 4 ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢𝐴𝑣𝐴)) → ((𝐹𝑢) ∈ 𝐵 ∧ (𝐹𝑣) ∈ 𝐵))
10 eqeq1 2062 . . . . . 6 (𝑥 = (𝐹𝑢) → (𝑥 = 𝑦 ↔ (𝐹𝑢) = 𝑦))
11 breq1 3795 . . . . . . . 8 (𝑥 = (𝐹𝑢) → (𝑥𝑆𝑦 ↔ (𝐹𝑢)𝑆𝑦))
1211notbid 602 . . . . . . 7 (𝑥 = (𝐹𝑢) → (¬ 𝑥𝑆𝑦 ↔ ¬ (𝐹𝑢)𝑆𝑦))
13 breq2 3796 . . . . . . . 8 (𝑥 = (𝐹𝑢) → (𝑦𝑆𝑥𝑦𝑆(𝐹𝑢)))
1413notbid 602 . . . . . . 7 (𝑥 = (𝐹𝑢) → (¬ 𝑦𝑆𝑥 ↔ ¬ 𝑦𝑆(𝐹𝑢)))
1512, 14anbi12d 450 . . . . . 6 (𝑥 = (𝐹𝑢) → ((¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥) ↔ (¬ (𝐹𝑢)𝑆𝑦 ∧ ¬ 𝑦𝑆(𝐹𝑢))))
1610, 15bibi12d 228 . . . . 5 (𝑥 = (𝐹𝑢) → ((𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥)) ↔ ((𝐹𝑢) = 𝑦 ↔ (¬ (𝐹𝑢)𝑆𝑦 ∧ ¬ 𝑦𝑆(𝐹𝑢)))))
17 eqeq2 2065 . . . . . 6 (𝑦 = (𝐹𝑣) → ((𝐹𝑢) = 𝑦 ↔ (𝐹𝑢) = (𝐹𝑣)))
18 breq2 3796 . . . . . . . 8 (𝑦 = (𝐹𝑣) → ((𝐹𝑢)𝑆𝑦 ↔ (𝐹𝑢)𝑆(𝐹𝑣)))
1918notbid 602 . . . . . . 7 (𝑦 = (𝐹𝑣) → (¬ (𝐹𝑢)𝑆𝑦 ↔ ¬ (𝐹𝑢)𝑆(𝐹𝑣)))
20 breq1 3795 . . . . . . . 8 (𝑦 = (𝐹𝑣) → (𝑦𝑆(𝐹𝑢) ↔ (𝐹𝑣)𝑆(𝐹𝑢)))
2120notbid 602 . . . . . . 7 (𝑦 = (𝐹𝑣) → (¬ 𝑦𝑆(𝐹𝑢) ↔ ¬ (𝐹𝑣)𝑆(𝐹𝑢)))
2219, 21anbi12d 450 . . . . . 6 (𝑦 = (𝐹𝑣) → ((¬ (𝐹𝑢)𝑆𝑦 ∧ ¬ 𝑦𝑆(𝐹𝑢)) ↔ (¬ (𝐹𝑢)𝑆(𝐹𝑣) ∧ ¬ (𝐹𝑣)𝑆(𝐹𝑢))))
2317, 22bibi12d 228 . . . . 5 (𝑦 = (𝐹𝑣) → (((𝐹𝑢) = 𝑦 ↔ (¬ (𝐹𝑢)𝑆𝑦 ∧ ¬ 𝑦𝑆(𝐹𝑢))) ↔ ((𝐹𝑢) = (𝐹𝑣) ↔ (¬ (𝐹𝑢)𝑆(𝐹𝑣) ∧ ¬ (𝐹𝑣)𝑆(𝐹𝑢)))))
2416, 23rspc2v 2685 . . . 4 (((𝐹𝑢) ∈ 𝐵 ∧ (𝐹𝑣) ∈ 𝐵) → (∀𝑥𝐵𝑦𝐵 (𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥)) → ((𝐹𝑢) = (𝐹𝑣) ↔ (¬ (𝐹𝑢)𝑆(𝐹𝑣) ∧ ¬ (𝐹𝑣)𝑆(𝐹𝑢)))))
259, 24syl 14 . . 3 ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢𝐴𝑣𝐴)) → (∀𝑥𝐵𝑦𝐵 (𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥)) → ((𝐹𝑢) = (𝐹𝑣) ↔ (¬ (𝐹𝑢)𝑆(𝐹𝑣) ∧ ¬ (𝐹𝑣)𝑆(𝐹𝑢)))))
26 f1of1 5153 . . . . . . 7 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴1-1𝐵)
271, 26syl 14 . . . . . 6 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴1-1𝐵)
28 f1fveq 5439 . . . . . 6 ((𝐹:𝐴1-1𝐵 ∧ (𝑢𝐴𝑣𝐴)) → ((𝐹𝑢) = (𝐹𝑣) ↔ 𝑢 = 𝑣))
2927, 28sylan 271 . . . . 5 ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢𝐴𝑣𝐴)) → ((𝐹𝑢) = (𝐹𝑣) ↔ 𝑢 = 𝑣))
3029bicomd 133 . . . 4 ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (𝐹𝑢) = (𝐹𝑣)))
31 isorel 5476 . . . . . 6 ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢𝐴𝑣𝐴)) → (𝑢𝑅𝑣 ↔ (𝐹𝑢)𝑆(𝐹𝑣)))
3231notbid 602 . . . . 5 ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢𝐴𝑣𝐴)) → (¬ 𝑢𝑅𝑣 ↔ ¬ (𝐹𝑢)𝑆(𝐹𝑣)))
33 isorel 5476 . . . . . . 7 ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑣𝐴𝑢𝐴)) → (𝑣𝑅𝑢 ↔ (𝐹𝑣)𝑆(𝐹𝑢)))
3433notbid 602 . . . . . 6 ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑣𝐴𝑢𝐴)) → (¬ 𝑣𝑅𝑢 ↔ ¬ (𝐹𝑣)𝑆(𝐹𝑢)))
3534ancom2s 508 . . . . 5 ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢𝐴𝑣𝐴)) → (¬ 𝑣𝑅𝑢 ↔ ¬ (𝐹𝑣)𝑆(𝐹𝑢)))
3632, 35anbi12d 450 . . . 4 ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢𝐴𝑣𝐴)) → ((¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢) ↔ (¬ (𝐹𝑢)𝑆(𝐹𝑣) ∧ ¬ (𝐹𝑣)𝑆(𝐹𝑢))))
3730, 36bibi12d 228 . . 3 ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢𝐴𝑣𝐴)) → ((𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)) ↔ ((𝐹𝑢) = (𝐹𝑣) ↔ (¬ (𝐹𝑢)𝑆(𝐹𝑣) ∧ ¬ (𝐹𝑣)𝑆(𝐹𝑢)))))
3825, 37sylibrd 162 . 2 ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢𝐴𝑣𝐴)) → (∀𝑥𝐵𝑦𝐵 (𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))))
3938ralrimdvva 2421 1 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑥𝐵𝑦𝐵 (𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥)) → ∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  wral 2323   class class class wbr 3792  wf 4926  1-1wf1 4927  1-1-ontowf1o 4929  cfv 4930   Isom wiso 4931
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-f1o 4937  df-fv 4938  df-isom 4939
This theorem is referenced by:  isoti  6411
  Copyright terms: Public domain W3C validator