Theorem isotilem 7072

Description: Lemma for isoti 7073. (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 5854	. . . . . 6 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴–1-1-onto→𝐵)
2		f1of 5504	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴⟶𝐵)
3		ffvelcdm 5695	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑢 ∈ 𝐴) → (𝐹‘𝑢) ∈ 𝐵)
4	3	ex 115	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → (𝑢 ∈ 𝐴 → (𝐹‘𝑢) ∈ 𝐵))
5		ffvelcdm 5695	. . . . . . . 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 2203	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑢) → (𝑥 = 𝑦 ↔ (𝐹‘𝑢) = 𝑦))
11		breq1 4036	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑢) → (𝑥𝑆𝑦 ↔ (𝐹‘𝑢)𝑆𝑦))
12	11	notbid 668	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑢) → (¬ 𝑥𝑆𝑦 ↔ ¬ (𝐹‘𝑢)𝑆𝑦))
13		breq2 4037	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑢) → (𝑦𝑆𝑥 ↔ 𝑦𝑆(𝐹‘𝑢)))
14	13	notbid 668	. . . . . . 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 2206	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝑣) → ((𝐹‘𝑢) = 𝑦 ↔ (𝐹‘𝑢) = (𝐹‘𝑣)))
18		breq2 4037	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑣) → ((𝐹‘𝑢)𝑆𝑦 ↔ (𝐹‘𝑢)𝑆(𝐹‘𝑣)))
19	18	notbid 668	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑣) → (¬ (𝐹‘𝑢)𝑆𝑦 ↔ ¬ (𝐹‘𝑢)𝑆(𝐹‘𝑣)))
20		breq1 4036	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑣) → (𝑦𝑆(𝐹‘𝑢) ↔ (𝐹‘𝑣)𝑆(𝐹‘𝑢)))
21	20	notbid 668	. . . . . . 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 2881	. . . 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 5503	. . . . . . 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 5819	. . . . . 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 5855	. . . . . 6 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢𝑅𝑣 ↔ (𝐹‘𝑢)𝑆(𝐹‘𝑣)))
32	31	notbid 668	. . . . 5 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (¬ 𝑢𝑅𝑣 ↔ ¬ (𝐹‘𝑢)𝑆(𝐹‘𝑣)))
33		isorel 5855	. . . . . . 7 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑣 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴)) → (𝑣𝑅𝑢 ↔ (𝐹‘𝑣)𝑆(𝐹‘𝑢)))
34	33	notbid 668	. . . . . 6 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑣 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴)) → (¬ 𝑣𝑅𝑢 ↔ ¬ (𝐹‘𝑣)𝑆(𝐹‘𝑢)))
35	34	ancom2s 566	. . . . 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 2582	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 1364   ∈ wcel 2167  ∀wral 2475   class class class wbr 4033  ⟶wf 5254  –1-1→wf1 5255  –1-1-onto→wf1o 5257  ‘cfv 5258   Isom wiso 5259

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-f1o 5265  df-fv 5266  df-isom 5267

This theorem is referenced by:  isoti  7073