Theorem isotilem 6999

Description: Lemma for isoti 7000. (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 5802	. . . . . 6 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴–1-1-onto→𝐵)
2		f1of 5457	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴⟶𝐵)
3		ffvelcdm 5645	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑢 ∈ 𝐴) → (𝐹‘𝑢) ∈ 𝐵)
4	3	ex 115	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → (𝑢 ∈ 𝐴 → (𝐹‘𝑢) ∈ 𝐵))
5		ffvelcdm 5645	. . . . . . . 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 2184	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑢) → (𝑥 = 𝑦 ↔ (𝐹‘𝑢) = 𝑦))
11		breq1 4003	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑢) → (𝑥𝑆𝑦 ↔ (𝐹‘𝑢)𝑆𝑦))
12	11	notbid 667	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑢) → (¬ 𝑥𝑆𝑦 ↔ ¬ (𝐹‘𝑢)𝑆𝑦))
13		breq2 4004	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑢) → (𝑦𝑆𝑥 ↔ 𝑦𝑆(𝐹‘𝑢)))
14	13	notbid 667	. . . . . . 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 2187	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝑣) → ((𝐹‘𝑢) = 𝑦 ↔ (𝐹‘𝑢) = (𝐹‘𝑣)))
18		breq2 4004	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑣) → ((𝐹‘𝑢)𝑆𝑦 ↔ (𝐹‘𝑢)𝑆(𝐹‘𝑣)))
19	18	notbid 667	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑣) → (¬ (𝐹‘𝑢)𝑆𝑦 ↔ ¬ (𝐹‘𝑢)𝑆(𝐹‘𝑣)))
20		breq1 4003	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑣) → (𝑦𝑆(𝐹‘𝑢) ↔ (𝐹‘𝑣)𝑆(𝐹‘𝑢)))
21	20	notbid 667	. . . . . . 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 2854	. . . 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 5456	. . . . . . 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 5767	. . . . . 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 5803	. . . . . 6 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢𝑅𝑣 ↔ (𝐹‘𝑢)𝑆(𝐹‘𝑣)))
32	31	notbid 667	. . . . 5 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (¬ 𝑢𝑅𝑣 ↔ ¬ (𝐹‘𝑢)𝑆(𝐹‘𝑣)))
33		isorel 5803	. . . . . . 7 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑣 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴)) → (𝑣𝑅𝑢 ↔ (𝐹‘𝑣)𝑆(𝐹‘𝑢)))
34	33	notbid 667	. . . . . 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 2562	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 1353   ∈ wcel 2148  ∀wral 2455   class class class wbr 4000  ⟶wf 5208  –1-1→wf1 5209  –1-1-onto→wf1o 5211  ‘cfv 5212   Isom wiso 5213

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-f1o 5219  df-fv 5220  df-isom 5221

This theorem is referenced by:  isoti  7000