Theorem isoti 7249

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 5962	. . . 4 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ◡𝐹 Isom 𝑆, 𝑅 (𝐵, 𝐴))
2		isotilem 7248	. . . 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 7248	. . 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 129	. 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 1760	. . . 4 ⊢ (𝑥 = 𝑢 → (𝑥 = 𝑦 ↔ 𝑢 = 𝑦))
7		breq1 4096	. . . . . 6 ⊢ (𝑥 = 𝑢 → (𝑥𝑆𝑦 ↔ 𝑢𝑆𝑦))
8	7	notbid 673	. . . . 5 ⊢ (𝑥 = 𝑢 → (¬ 𝑥𝑆𝑦 ↔ ¬ 𝑢𝑆𝑦))
9		breq2 4097	. . . . . 6 ⊢ (𝑥 = 𝑢 → (𝑦𝑆𝑥 ↔ 𝑦𝑆𝑢))
10	9	notbid 673	. . . . 5 ⊢ (𝑥 = 𝑢 → (¬ 𝑦𝑆𝑥 ↔ ¬ 𝑦𝑆𝑢))
11	8, 10	anbi12d 473	. . . 4 ⊢ (𝑥 = 𝑢 → ((¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥) ↔ (¬ 𝑢𝑆𝑦 ∧ ¬ 𝑦𝑆𝑢)))
12	6, 11	bibi12d 235	. . 3 ⊢ (𝑥 = 𝑢 → ((𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥)) ↔ (𝑢 = 𝑦 ↔ (¬ 𝑢𝑆𝑦 ∧ ¬ 𝑦𝑆𝑢))))
13		equequ2 1761	. . . 4 ⊢ (𝑦 = 𝑣 → (𝑢 = 𝑦 ↔ 𝑢 = 𝑣))
14		breq2 4097	. . . . . 6 ⊢ (𝑦 = 𝑣 → (𝑢𝑆𝑦 ↔ 𝑢𝑆𝑣))
15	14	notbid 673	. . . . 5 ⊢ (𝑦 = 𝑣 → (¬ 𝑢𝑆𝑦 ↔ ¬ 𝑢𝑆𝑣))
16		breq1 4096	. . . . . 6 ⊢ (𝑦 = 𝑣 → (𝑦𝑆𝑢 ↔ 𝑣𝑆𝑢))
17	16	notbid 673	. . . . 5 ⊢ (𝑦 = 𝑣 → (¬ 𝑦𝑆𝑢 ↔ ¬ 𝑣𝑆𝑢))
18	15, 17	anbi12d 473	. . . 4 ⊢ (𝑦 = 𝑣 → ((¬ 𝑢𝑆𝑦 ∧ ¬ 𝑦𝑆𝑢) ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢)))
19	13, 18	bibi12d 235	. . 3 ⊢ (𝑦 = 𝑣 → ((𝑢 = 𝑦 ↔ (¬ 𝑢𝑆𝑦 ∧ ¬ 𝑦𝑆𝑢)) ↔ (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢))))
20	12, 19	cbvral2v 2781	. 2 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥)) ↔ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢)))
21	5, 20	bitrdi 196	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 104   ↔ wb 105  ∀wral 2511   class class class wbr 4093  ^◡ccnv 4730   Isom wiso 5334

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-isom 5342

This theorem is referenced by:  supisoti  7252