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) → (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)) ↔ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (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.

Step	Hyp	Ref	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) → (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥))))
3	1, 2	syl 14	. . 3 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥))))
4		isotilem 6942	. . 3 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥)) → ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))))
5	3, 4	impbid 128	. 2 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥))))
6		equequ1 1692	. . . 4 ⊢ (𝑥 = 𝑢 → (𝑥 = 𝑦 ↔ 𝑢 = 𝑦))
7		breq1 3968	. . . . . 6 ⊢ (𝑥 = 𝑢 → (𝑥𝑆𝑦 ↔ 𝑢𝑆𝑦))
8	7	notbid 657	. . . . 5 ⊢ (𝑥 = 𝑢 → (¬ 𝑥𝑆𝑦 ↔ ¬ 𝑢𝑆𝑦))
9		breq2 3969	. . . . . 6 ⊢ (𝑥 = 𝑢 → (𝑦𝑆𝑥 ↔ 𝑦𝑆𝑢))
10	9	notbid 657	. . . . 5 ⊢ (𝑥 = 𝑢 → (¬ 𝑦𝑆𝑥 ↔ ¬ 𝑦𝑆𝑢))
11	8, 10	anbi12d 465	. . . 4 ⊢ (𝑥 = 𝑢 → ((¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥) ↔ (¬ 𝑢𝑆𝑦 ∧ ¬ 𝑦𝑆𝑢)))
12	6, 11	bibi12d 234	. . 3 ⊢ (𝑥 = 𝑢 → ((𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥)) ↔ (𝑢 = 𝑦 ↔ (¬ 𝑢𝑆𝑦 ∧ ¬ 𝑦𝑆𝑢))))
13		equequ2 1693	. . . 4 ⊢ (𝑦 = 𝑣 → (𝑢 = 𝑦 ↔ 𝑢 = 𝑣))
14		breq2 3969	. . . . . 6 ⊢ (𝑦 = 𝑣 → (𝑢𝑆𝑦 ↔ 𝑢𝑆𝑣))
15	14	notbid 657	. . . . 5 ⊢ (𝑦 = 𝑣 → (¬ 𝑢𝑆𝑦 ↔ ¬ 𝑢𝑆𝑣))
16		breq1 3968	. . . . . 6 ⊢ (𝑦 = 𝑣 → (𝑦𝑆𝑢 ↔ 𝑣𝑆𝑢))
17	16	notbid 657	. . . . 5 ⊢ (𝑦 = 𝑣 → (¬ 𝑦𝑆𝑢 ↔ ¬ 𝑣𝑆𝑢))
18	15, 17	anbi12d 465	. . . 4 ⊢ (𝑦 = 𝑣 → ((¬ 𝑢𝑆𝑦 ∧ ¬ 𝑦𝑆𝑢) ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢)))
19	13, 18	bibi12d 234	. . 3 ⊢ (𝑦 = 𝑣 → ((𝑢 = 𝑦 ↔ (¬ 𝑢𝑆𝑦 ∧ ¬ 𝑦𝑆𝑢)) ↔ (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢))))
20	12, 19	cbvral2v 2691	. 2 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥)) ↔ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢)))
21	5, 20	bitrdi 195	1 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)) ↔ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢))))

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