Theorem isoti 6758

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 5606	. . . 4 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ◡𝐹 Isom 𝑆, 𝑅 (𝐵, 𝐴))
2		isotilem 6757	. . . 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 6757	. . 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 1646	. . . 4 ⊢ (𝑥 = 𝑢 → (𝑥 = 𝑦 ↔ 𝑢 = 𝑦))
7		breq1 3856	. . . . . 6 ⊢ (𝑥 = 𝑢 → (𝑥𝑆𝑦 ↔ 𝑢𝑆𝑦))
8	7	notbid 628	. . . . 5 ⊢ (𝑥 = 𝑢 → (¬ 𝑥𝑆𝑦 ↔ ¬ 𝑢𝑆𝑦))
9		breq2 3857	. . . . . 6 ⊢ (𝑥 = 𝑢 → (𝑦𝑆𝑥 ↔ 𝑦𝑆𝑢))
10	9	notbid 628	. . . . 5 ⊢ (𝑥 = 𝑢 → (¬ 𝑦𝑆𝑥 ↔ ¬ 𝑦𝑆𝑢))
11	8, 10	anbi12d 458	. . . 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 1647	. . . 4 ⊢ (𝑦 = 𝑣 → (𝑢 = 𝑦 ↔ 𝑢 = 𝑣))
14		breq2 3857	. . . . . 6 ⊢ (𝑦 = 𝑣 → (𝑢𝑆𝑦 ↔ 𝑢𝑆𝑣))
15	14	notbid 628	. . . . 5 ⊢ (𝑦 = 𝑣 → (¬ 𝑢𝑆𝑦 ↔ ¬ 𝑢𝑆𝑣))
16		breq1 3856	. . . . . 6 ⊢ (𝑦 = 𝑣 → (𝑦𝑆𝑢 ↔ 𝑣𝑆𝑢))
17	16	notbid 628	. . . . 5 ⊢ (𝑦 = 𝑣 → (¬ 𝑦𝑆𝑢 ↔ ¬ 𝑣𝑆𝑢))
18	15, 17	anbi12d 458	. . . 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 2601	. 2 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥)) ↔ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢)))
21	5, 20	syl6bb 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 2360   class class class wbr 3853  ^◡ccnv 4453   Isom wiso 5031

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2624  df-sbc 2844  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-br 3854  df-opab 3908  df-id 4131  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-rn 4465  df-res 4466  df-ima 4467  df-iota 4995  df-fun 5032  df-fn 5033  df-f 5034  df-f1 5035  df-fo 5036  df-f1o 5037  df-fv 5038  df-isom 5039

This theorem is referenced by:  supisoti  6761