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Theorem isotilem 6859
 Description: Lemma for isoti 6860. (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 5674 . . . . . 6 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴1-1-onto𝐵)
2 f1of 5333 . . . . . 6 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)
3 ffvelrn 5519 . . . . . . . 8 ((𝐹:𝐴𝐵𝑢𝐴) → (𝐹𝑢) ∈ 𝐵)
43ex 114 . . . . . . 7 (𝐹:𝐴𝐵 → (𝑢𝐴 → (𝐹𝑢) ∈ 𝐵))
5 ffvelrn 5519 . . . . . . . 8 ((𝐹:𝐴𝐵𝑣𝐴) → (𝐹𝑣) ∈ 𝐵)
65ex 114 . . . . . . 7 (𝐹:𝐴𝐵 → (𝑣𝐴 → (𝐹𝑣) ∈ 𝐵))
74, 6anim12d 331 . . . . . 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 123 . . . 4 ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢𝐴𝑣𝐴)) → ((𝐹𝑢) ∈ 𝐵 ∧ (𝐹𝑣) ∈ 𝐵))
10 eqeq1 2122 . . . . . 6 (𝑥 = (𝐹𝑢) → (𝑥 = 𝑦 ↔ (𝐹𝑢) = 𝑦))
11 breq1 3900 . . . . . . . 8 (𝑥 = (𝐹𝑢) → (𝑥𝑆𝑦 ↔ (𝐹𝑢)𝑆𝑦))
1211notbid 639 . . . . . . 7 (𝑥 = (𝐹𝑢) → (¬ 𝑥𝑆𝑦 ↔ ¬ (𝐹𝑢)𝑆𝑦))
13 breq2 3901 . . . . . . . 8 (𝑥 = (𝐹𝑢) → (𝑦𝑆𝑥𝑦𝑆(𝐹𝑢)))
1413notbid 639 . . . . . . 7 (𝑥 = (𝐹𝑢) → (¬ 𝑦𝑆𝑥 ↔ ¬ 𝑦𝑆(𝐹𝑢)))
1512, 14anbi12d 462 . . . . . 6 (𝑥 = (𝐹𝑢) → ((¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥) ↔ (¬ (𝐹𝑢)𝑆𝑦 ∧ ¬ 𝑦𝑆(𝐹𝑢))))
1610, 15bibi12d 234 . . . . 5 (𝑥 = (𝐹𝑢) → ((𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥)) ↔ ((𝐹𝑢) = 𝑦 ↔ (¬ (𝐹𝑢)𝑆𝑦 ∧ ¬ 𝑦𝑆(𝐹𝑢)))))
17 eqeq2 2125 . . . . . 6 (𝑦 = (𝐹𝑣) → ((𝐹𝑢) = 𝑦 ↔ (𝐹𝑢) = (𝐹𝑣)))
18 breq2 3901 . . . . . . . 8 (𝑦 = (𝐹𝑣) → ((𝐹𝑢)𝑆𝑦 ↔ (𝐹𝑢)𝑆(𝐹𝑣)))
1918notbid 639 . . . . . . 7 (𝑦 = (𝐹𝑣) → (¬ (𝐹𝑢)𝑆𝑦 ↔ ¬ (𝐹𝑢)𝑆(𝐹𝑣)))
20 breq1 3900 . . . . . . . 8 (𝑦 = (𝐹𝑣) → (𝑦𝑆(𝐹𝑢) ↔ (𝐹𝑣)𝑆(𝐹𝑢)))
2120notbid 639 . . . . . . 7 (𝑦 = (𝐹𝑣) → (¬ 𝑦𝑆(𝐹𝑢) ↔ ¬ (𝐹𝑣)𝑆(𝐹𝑢)))
2219, 21anbi12d 462 . . . . . 6 (𝑦 = (𝐹𝑣) → ((¬ (𝐹𝑢)𝑆𝑦 ∧ ¬ 𝑦𝑆(𝐹𝑢)) ↔ (¬ (𝐹𝑢)𝑆(𝐹𝑣) ∧ ¬ (𝐹𝑣)𝑆(𝐹𝑢))))
2317, 22bibi12d 234 . . . . 5 (𝑦 = (𝐹𝑣) → (((𝐹𝑢) = 𝑦 ↔ (¬ (𝐹𝑢)𝑆𝑦 ∧ ¬ 𝑦𝑆(𝐹𝑢))) ↔ ((𝐹𝑢) = (𝐹𝑣) ↔ (¬ (𝐹𝑢)𝑆(𝐹𝑣) ∧ ¬ (𝐹𝑣)𝑆(𝐹𝑢)))))
2416, 23rspc2v 2774 . . . 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 5332 . . . . . . 7 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴1-1𝐵)
271, 26syl 14 . . . . . 6 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴1-1𝐵)
28 f1fveq 5639 . . . . . 6 ((𝐹:𝐴1-1𝐵 ∧ (𝑢𝐴𝑣𝐴)) → ((𝐹𝑢) = (𝐹𝑣) ↔ 𝑢 = 𝑣))
2927, 28sylan 279 . . . . 5 ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢𝐴𝑣𝐴)) → ((𝐹𝑢) = (𝐹𝑣) ↔ 𝑢 = 𝑣))
3029bicomd 140 . . . 4 ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (𝐹𝑢) = (𝐹𝑣)))
31 isorel 5675 . . . . . 6 ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢𝐴𝑣𝐴)) → (𝑢𝑅𝑣 ↔ (𝐹𝑢)𝑆(𝐹𝑣)))
3231notbid 639 . . . . 5 ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢𝐴𝑣𝐴)) → (¬ 𝑢𝑅𝑣 ↔ ¬ (𝐹𝑢)𝑆(𝐹𝑣)))
33 isorel 5675 . . . . . . 7 ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑣𝐴𝑢𝐴)) → (𝑣𝑅𝑢 ↔ (𝐹𝑣)𝑆(𝐹𝑢)))
3433notbid 639 . . . . . 6 ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑣𝐴𝑢𝐴)) → (¬ 𝑣𝑅𝑢 ↔ ¬ (𝐹𝑣)𝑆(𝐹𝑢)))
3534ancom2s 538 . . . . 5 ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢𝐴𝑣𝐴)) → (¬ 𝑣𝑅𝑢 ↔ ¬ (𝐹𝑣)𝑆(𝐹𝑢)))
3632, 35anbi12d 462 . . . 4 ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢𝐴𝑣𝐴)) → ((¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢) ↔ (¬ (𝐹𝑢)𝑆(𝐹𝑣) ∧ ¬ (𝐹𝑣)𝑆(𝐹𝑢))))
3730, 36bibi12d 234 . . 3 ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢𝐴𝑣𝐴)) → ((𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)) ↔ ((𝐹𝑢) = (𝐹𝑣) ↔ (¬ (𝐹𝑢)𝑆(𝐹𝑣) ∧ ¬ (𝐹𝑣)𝑆(𝐹𝑢)))))
3825, 37sylibrd 168 . 2 ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢𝐴𝑣𝐴)) → (∀𝑥𝐵𝑦𝐵 (𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))))
3938ralrimdvva 2492 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 103   ↔ wb 104   = wceq 1314   ∈ wcel 1463  ∀wral 2391   class class class wbr 3897  ⟶wf 5087  –1-1→wf1 5088  –1-1-onto→wf1o 5090  ‘cfv 5091   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-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-f1o 5098  df-fv 5099  df-isom 5100 This theorem is referenced by:  isoti  6860
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