Theorem isotilem 6886

Description: Lemma for isoti 6887. (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 5701	. . . . . 6 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴–1-1-onto→𝐵)
2		f1of 5360	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴⟶𝐵)
3		ffvelrn 5546	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑢 ∈ 𝐴) → (𝐹‘𝑢) ∈ 𝐵)
4	3	ex 114	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → (𝑢 ∈ 𝐴 → (𝐹‘𝑢) ∈ 𝐵))
5		ffvelrn 5546	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑣 ∈ 𝐴) → (𝐹‘𝑣) ∈ 𝐵)
6	5	ex 114	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → (𝑣 ∈ 𝐴 → (𝐹‘𝑣) ∈ 𝐵))
7	4, 6	anim12d 333	. . . . . 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 123	. . . 4 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → ((𝐹‘𝑢) ∈ 𝐵 ∧ (𝐹‘𝑣) ∈ 𝐵))
10		eqeq1 2144	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑢) → (𝑥 = 𝑦 ↔ (𝐹‘𝑢) = 𝑦))
11		breq1 3927	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑢) → (𝑥𝑆𝑦 ↔ (𝐹‘𝑢)𝑆𝑦))
12	11	notbid 656	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑢) → (¬ 𝑥𝑆𝑦 ↔ ¬ (𝐹‘𝑢)𝑆𝑦))
13		breq2 3928	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑢) → (𝑦𝑆𝑥 ↔ 𝑦𝑆(𝐹‘𝑢)))
14	13	notbid 656	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑢) → (¬ 𝑦𝑆𝑥 ↔ ¬ 𝑦𝑆(𝐹‘𝑢)))
15	12, 14	anbi12d 464	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑢) → ((¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥) ↔ (¬ (𝐹‘𝑢)𝑆𝑦 ∧ ¬ 𝑦𝑆(𝐹‘𝑢))))
16	10, 15	bibi12d 234	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑢) → ((𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥)) ↔ ((𝐹‘𝑢) = 𝑦 ↔ (¬ (𝐹‘𝑢)𝑆𝑦 ∧ ¬ 𝑦𝑆(𝐹‘𝑢)))))
17		eqeq2 2147	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝑣) → ((𝐹‘𝑢) = 𝑦 ↔ (𝐹‘𝑢) = (𝐹‘𝑣)))
18		breq2 3928	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑣) → ((𝐹‘𝑢)𝑆𝑦 ↔ (𝐹‘𝑢)𝑆(𝐹‘𝑣)))
19	18	notbid 656	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑣) → (¬ (𝐹‘𝑢)𝑆𝑦 ↔ ¬ (𝐹‘𝑢)𝑆(𝐹‘𝑣)))
20		breq1 3927	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑣) → (𝑦𝑆(𝐹‘𝑢) ↔ (𝐹‘𝑣)𝑆(𝐹‘𝑢)))
21	20	notbid 656	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑣) → (¬ 𝑦𝑆(𝐹‘𝑢) ↔ ¬ (𝐹‘𝑣)𝑆(𝐹‘𝑢)))
22	19, 21	anbi12d 464	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝑣) → ((¬ (𝐹‘𝑢)𝑆𝑦 ∧ ¬ 𝑦𝑆(𝐹‘𝑢)) ↔ (¬ (𝐹‘𝑢)𝑆(𝐹‘𝑣) ∧ ¬ (𝐹‘𝑣)𝑆(𝐹‘𝑢))))
23	17, 22	bibi12d 234	. . . . 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 2797	. . . 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 5359	. . . . . . 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 5666	. . . . . 6 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ 𝑢 = 𝑣))
29	27, 28	sylan 281	. . . . 5 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ 𝑢 = 𝑣))
30	29	bicomd 140	. . . 4 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (𝐹‘𝑢) = (𝐹‘𝑣)))
31		isorel 5702	. . . . . 6 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢𝑅𝑣 ↔ (𝐹‘𝑢)𝑆(𝐹‘𝑣)))
32	31	notbid 656	. . . . 5 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (¬ 𝑢𝑅𝑣 ↔ ¬ (𝐹‘𝑢)𝑆(𝐹‘𝑣)))
33		isorel 5702	. . . . . . 7 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑣 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴)) → (𝑣𝑅𝑢 ↔ (𝐹‘𝑣)𝑆(𝐹‘𝑢)))
34	33	notbid 656	. . . . . 6 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑣 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴)) → (¬ 𝑣𝑅𝑢 ↔ ¬ (𝐹‘𝑣)𝑆(𝐹‘𝑢)))
35	34	ancom2s 555	. . . . 5 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (¬ 𝑣𝑅𝑢 ↔ ¬ (𝐹‘𝑣)𝑆(𝐹‘𝑢)))
36	32, 35	anbi12d 464	. . . 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 234	. . 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 168	. 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 2515	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 103   ↔ wb 104   = wceq 1331   ∈ wcel 1480  ∀wral 2414   class class class wbr 3924  ⟶wf 5114  –1-1→wf1 5115  –1-1-onto→wf1o 5117  ‘cfv 5118   Isom wiso 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-f1o 5125  df-fv 5126  df-isom 5127

This theorem is referenced by:  isoti  6887