Theorem isotilem 7204

Description: Lemma for isoti 7205. (Contributed by Jim Kingdon, 26-Nov-2021.)

Assertion

Ref	Expression
isotilem	⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥)) → ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (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

Step	Hyp	Ref	Expression
1		isof1o 5947	. . . . . 6 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴–1-1-onto→𝐵)
2		f1of 5583	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴⟶𝐵)
3		ffvelcdm 5780	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑢 ∈ 𝐴) → (𝐹‘𝑢) ∈ 𝐵)
4	3	ex 115	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → (𝑢 ∈ 𝐴 → (𝐹‘𝑢) ∈ 𝐵))
5		ffvelcdm 5780	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑣 ∈ 𝐴) → (𝐹‘𝑣) ∈ 𝐵)
6	5	ex 115	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → (𝑣 ∈ 𝐴 → (𝐹‘𝑣) ∈ 𝐵))
7	4, 6	anim12d 335	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → ((𝐹‘𝑢) ∈ 𝐵 ∧ (𝐹‘𝑣) ∈ 𝐵)))
8	1, 2, 7	3syl 17	. . . . 5 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → ((𝐹‘𝑢) ∈ 𝐵 ∧ (𝐹‘𝑣) ∈ 𝐵)))
9	8	imp 124	. . . 4 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → ((𝐹‘𝑢) ∈ 𝐵 ∧ (𝐹‘𝑣) ∈ 𝐵))
10		eqeq1 2238	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑢) → (𝑥 = 𝑦 ↔ (𝐹‘𝑢) = 𝑦))
11		breq1 4091	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑢) → (𝑥𝑆𝑦 ↔ (𝐹‘𝑢)𝑆𝑦))
12	11	notbid 673	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑢) → (¬ 𝑥𝑆𝑦 ↔ ¬ (𝐹‘𝑢)𝑆𝑦))
13		breq2 4092	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑢) → (𝑦𝑆𝑥 ↔ 𝑦𝑆(𝐹‘𝑢)))
14	13	notbid 673	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑢) → (¬ 𝑦𝑆𝑥 ↔ ¬ 𝑦𝑆(𝐹‘𝑢)))
15	12, 14	anbi12d 473	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑢) → ((¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥) ↔ (¬ (𝐹‘𝑢)𝑆𝑦 ∧ ¬ 𝑦𝑆(𝐹‘𝑢))))
16	10, 15	bibi12d 235	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑢) → ((𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥)) ↔ ((𝐹‘𝑢) = 𝑦 ↔ (¬ (𝐹‘𝑢)𝑆𝑦 ∧ ¬ 𝑦𝑆(𝐹‘𝑢)))))
17		eqeq2 2241	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝑣) → ((𝐹‘𝑢) = 𝑦 ↔ (𝐹‘𝑢) = (𝐹‘𝑣)))
18		breq2 4092	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑣) → ((𝐹‘𝑢)𝑆𝑦 ↔ (𝐹‘𝑢)𝑆(𝐹‘𝑣)))
19	18	notbid 673	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑣) → (¬ (𝐹‘𝑢)𝑆𝑦 ↔ ¬ (𝐹‘𝑢)𝑆(𝐹‘𝑣)))
20		breq1 4091	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑣) → (𝑦𝑆(𝐹‘𝑢) ↔ (𝐹‘𝑣)𝑆(𝐹‘𝑢)))
21	20	notbid 673	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑣) → (¬ 𝑦𝑆(𝐹‘𝑢) ↔ ¬ (𝐹‘𝑣)𝑆(𝐹‘𝑢)))
22	19, 21	anbi12d 473	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝑣) → ((¬ (𝐹‘𝑢)𝑆𝑦 ∧ ¬ 𝑦𝑆(𝐹‘𝑢)) ↔ (¬ (𝐹‘𝑢)𝑆(𝐹‘𝑣) ∧ ¬ (𝐹‘𝑣)𝑆(𝐹‘𝑢))))
23	17, 22	bibi12d 235	. . . . 5 ⊢ (𝑦 = (𝐹‘𝑣) → (((𝐹‘𝑢) = 𝑦 ↔ (¬ (𝐹‘𝑢)𝑆𝑦 ∧ ¬ 𝑦𝑆(𝐹‘𝑢))) ↔ ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ (¬ (𝐹‘𝑢)𝑆(𝐹‘𝑣) ∧ ¬ (𝐹‘𝑣)𝑆(𝐹‘𝑢)))))
24	16, 23	rspc2v 2923	. . . 4 ⊢ (((𝐹‘𝑢) ∈ 𝐵 ∧ (𝐹‘𝑣) ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥)) → ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ (¬ (𝐹‘𝑢)𝑆(𝐹‘𝑣) ∧ ¬ (𝐹‘𝑣)𝑆(𝐹‘𝑢)))))
25	9, 24	syl 14	. . 3 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥)) → ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ (¬ (𝐹‘𝑢)𝑆(𝐹‘𝑣) ∧ ¬ (𝐹‘𝑣)𝑆(𝐹‘𝑢)))))
26		f1of1 5582	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–1-1→𝐵)
27	1, 26	syl 14	. . . . . 6 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴–1-1→𝐵)
28		f1fveq 5912	. . . . . 6 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ 𝑢 = 𝑣))
29	27, 28	sylan 283	. . . . 5 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ 𝑢 = 𝑣))
30	29	bicomd 141	. . . 4 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (𝐹‘𝑢) = (𝐹‘𝑣)))
31		isorel 5948	. . . . . 6 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢𝑅𝑣 ↔ (𝐹‘𝑢)𝑆(𝐹‘𝑣)))
32	31	notbid 673	. . . . 5 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (¬ 𝑢𝑅𝑣 ↔ ¬ (𝐹‘𝑢)𝑆(𝐹‘𝑣)))
33		isorel 5948	. . . . . . 7 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑣 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴)) → (𝑣𝑅𝑢 ↔ (𝐹‘𝑣)𝑆(𝐹‘𝑢)))
34	33	notbid 673	. . . . . 6 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑣 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴)) → (¬ 𝑣𝑅𝑢 ↔ ¬ (𝐹‘𝑣)𝑆(𝐹‘𝑢)))
35	34	ancom2s 568	. . . . 5 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (¬ 𝑣𝑅𝑢 ↔ ¬ (𝐹‘𝑣)𝑆(𝐹‘𝑢)))
36	32, 35	anbi12d 473	. . . 4 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → ((¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢) ↔ (¬ (𝐹‘𝑢)𝑆(𝐹‘𝑣) ∧ ¬ (𝐹‘𝑣)𝑆(𝐹‘𝑢))))
37	30, 36	bibi12d 235	. . 3 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → ((𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)) ↔ ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ (¬ (𝐹‘𝑢)𝑆(𝐹‘𝑣) ∧ ¬ (𝐹‘𝑣)𝑆(𝐹‘𝑢)))))
38	25, 37	sylibrd 169	. 2 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))))
39	38	ralrimdvva 2617	1 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥)) → ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1397   ∈ wcel 2202  ∀wral 2510   class class class wbr 4088  ⟶wf 5322  –1-1→wf1 5323  –1-1-onto→wf1o 5325  ‘cfv 5326   Isom wiso 5327

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-f1o 5333  df-fv 5334  df-isom 5335

This theorem is referenced by:  isoti  7205