Theorem isosolem 7354

Description: Lemma for isoso 7355. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Assertion

Ref	Expression
isosolem	⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Or 𝐵 → 𝑅 Or 𝐴))

Proof of Theorem isosolem

Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isopolem 7352	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Po 𝐵 → 𝑅 Po 𝐴))
2		isof1o 7330	. . . . . . . 8 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
3		f1of 6824	. . . . . . . 8 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴⟶𝐵)
4		ffvelcdm 7080	. . . . . . . . . 10 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑐 ∈ 𝐴) → (𝐻‘𝑐) ∈ 𝐵)
5	4	ex 418	. . . . . . . . 9 ⊢ (𝐻:𝐴⟶𝐵 → (𝑐 ∈ 𝐴 → (𝐻‘𝑐) ∈ 𝐵))
6		ffvelcdm 7080	. . . . . . . . . 10 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑑 ∈ 𝐴) → (𝐻‘𝑑) ∈ 𝐵)
7	6	ex 418	. . . . . . . . 9 ⊢ (𝐻:𝐴⟶𝐵 → (𝑑 ∈ 𝐴 → (𝐻‘𝑑) ∈ 𝐵))
8	5, 7	anim12d 621	. . . . . . . 8 ⊢ (𝐻:𝐴⟶𝐵 → ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) → ((𝐻‘𝑐) ∈ 𝐵 ∧ (𝐻‘𝑑) ∈ 𝐵)))
9	2, 3, 8	3syl 19	. . . . . . 7 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) → ((𝐻‘𝑐) ∈ 𝐵 ∧ (𝐻‘𝑑) ∈ 𝐵)))
10	9	imp 412	. . . . . 6 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) → ((𝐻‘𝑐) ∈ 𝐵 ∧ (𝐻‘𝑑) ∈ 𝐵))
11		breq1 5114	. . . . . . . 8 ⊢ (𝑎 = (𝐻‘𝑐) → (𝑎𝑆𝑏 ↔ (𝐻‘𝑐)𝑆𝑏))
12		eqeq1 2769	. . . . . . . 8 ⊢ (𝑎 = (𝐻‘𝑐) → (𝑎 = 𝑏 ↔ (𝐻‘𝑐) = 𝑏))
13		breq2 5115	. . . . . . . 8 ⊢ (𝑎 = (𝐻‘𝑐) → (𝑏𝑆𝑎 ↔ 𝑏𝑆(𝐻‘𝑐)))
14	11, 12, 13	3orbi123d 1463	. . . . . . 7 ⊢ (𝑎 = (𝐻‘𝑐) → ((𝑎𝑆𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑆𝑎) ↔ ((𝐻‘𝑐)𝑆𝑏 ∨ (𝐻‘𝑐) = 𝑏 ∨ 𝑏𝑆(𝐻‘𝑐))))
15		breq2 5115	. . . . . . . 8 ⊢ (𝑏 = (𝐻‘𝑑) → ((𝐻‘𝑐)𝑆𝑏 ↔ (𝐻‘𝑐)𝑆(𝐻‘𝑑)))
16		eqeq2 2777	. . . . . . . 8 ⊢ (𝑏 = (𝐻‘𝑑) → ((𝐻‘𝑐) = 𝑏 ↔ (𝐻‘𝑐) = (𝐻‘𝑑)))
17		breq1 5114	. . . . . . . 8 ⊢ (𝑏 = (𝐻‘𝑑) → (𝑏𝑆(𝐻‘𝑐) ↔ (𝐻‘𝑑)𝑆(𝐻‘𝑐)))
18	15, 16, 17	3orbi123d 1463	. . . . . . 7 ⊢ (𝑏 = (𝐻‘𝑑) → (((𝐻‘𝑐)𝑆𝑏 ∨ (𝐻‘𝑐) = 𝑏 ∨ 𝑏𝑆(𝐻‘𝑐)) ↔ ((𝐻‘𝑐)𝑆(𝐻‘𝑑) ∨ (𝐻‘𝑐) = (𝐻‘𝑑) ∨ (𝐻‘𝑑)𝑆(𝐻‘𝑐))))
19	14, 18	rspc2v 3594	. . . . . 6 ⊢ (((𝐻‘𝑐) ∈ 𝐵 ∧ (𝐻‘𝑑) ∈ 𝐵) → (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎𝑆𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑆𝑎) → ((𝐻‘𝑐)𝑆(𝐻‘𝑑) ∨ (𝐻‘𝑐) = (𝐻‘𝑑) ∨ (𝐻‘𝑑)𝑆(𝐻‘𝑐))))
20	10, 19	syl 18	. . . . 5 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) → (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎𝑆𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑆𝑎) → ((𝐻‘𝑐)𝑆(𝐻‘𝑑) ∨ (𝐻‘𝑐) = (𝐻‘𝑑) ∨ (𝐻‘𝑑)𝑆(𝐻‘𝑐))))
21		isorel 7333	. . . . . 6 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) → (𝑐𝑅𝑑 ↔ (𝐻‘𝑐)𝑆(𝐻‘𝑑)))
22		f1of1 6823	. . . . . . . . 9 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴–1-1→𝐵)
23	2, 22	syl 18	. . . . . . . 8 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1→𝐵)
24		f1fveq 7265	. . . . . . . 8 ⊢ ((𝐻:𝐴–1-1→𝐵 ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) → ((𝐻‘𝑐) = (𝐻‘𝑑) ↔ 𝑐 = 𝑑))
25	23, 24	sylan 592	. . . . . . 7 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) → ((𝐻‘𝑐) = (𝐻‘𝑑) ↔ 𝑐 = 𝑑))
26	25	bicomd 226	. . . . . 6 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) → (𝑐 = 𝑑 ↔ (𝐻‘𝑐) = (𝐻‘𝑑)))
27		isorel 7333	. . . . . . 7 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) → (𝑑𝑅𝑐 ↔ (𝐻‘𝑑)𝑆(𝐻‘𝑐)))
28	27	ancom2s 663	. . . . . 6 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) → (𝑑𝑅𝑐 ↔ (𝐻‘𝑑)𝑆(𝐻‘𝑐)))
29	21, 26, 28	3orbi123d 1463	. . . . 5 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) → ((𝑐𝑅𝑑 ∨ 𝑐 = 𝑑 ∨ 𝑑𝑅𝑐) ↔ ((𝐻‘𝑐)𝑆(𝐻‘𝑑) ∨ (𝐻‘𝑐) = (𝐻‘𝑑) ∨ (𝐻‘𝑑)𝑆(𝐻‘𝑐))))
30	20, 29	sylibrd 262	. . . 4 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) → (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎𝑆𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑆𝑎) → (𝑐𝑅𝑑 ∨ 𝑐 = 𝑑 ∨ 𝑑𝑅𝑐)))
31	30	ralrimdvva 3222	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎𝑆𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑆𝑎) → ∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐴 (𝑐𝑅𝑑 ∨ 𝑐 = 𝑑 ∨ 𝑑𝑅𝑐)))
32	1, 31	anim12d 621	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ((𝑆 Po 𝐵 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎𝑆𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑆𝑎)) → (𝑅 Po 𝐴 ∧ ∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐴 (𝑐𝑅𝑑 ∨ 𝑐 = 𝑑 ∨ 𝑑𝑅𝑐))))
33		df-so 5572	. 2 ⊢ (𝑆 Or 𝐵 ↔ (𝑆 Po 𝐵 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎𝑆𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑆𝑎)))
34		df-so 5572	. 2 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐴 (𝑐𝑅𝑑 ∨ 𝑐 = 𝑑 ∨ 𝑑𝑅𝑐)))
35	32, 33, 34	3imtr4g 299	1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Or 𝐵 → 𝑅 Or 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ w3o 1102   = wceq 1570   ∈ wcel 2146  ∀wral 3081   class class class wbr 5111   Po wpo 5569   Or wor 5570  ⟶wf 6536  –1-1→wf1 6537  –1-1-onto→wf1o 6539  ‘cfv 6540   Isom wiso 6541

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-id 5558  df-po 5571  df-so 5572  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-f1o 6547  df-fv 6548  df-isom 6549

This theorem is used by:  isoso  7355  isowe2  7357