Theorem isosolem 7284

Description: Lemma for isoso 7285. (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 7282	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Po 𝐵 → 𝑅 Po 𝐴))
2		isof1o 7260	. . . . . . . 8 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
3		f1of 6764	. . . . . . . 8 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴⟶𝐵)
4		ffvelcdm 7015	. . . . . . . . . 10 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑐 ∈ 𝐴) → (𝐻‘𝑐) ∈ 𝐵)
5	4	ex 412	. . . . . . . . 9 ⊢ (𝐻:𝐴⟶𝐵 → (𝑐 ∈ 𝐴 → (𝐻‘𝑐) ∈ 𝐵))
6		ffvelcdm 7015	. . . . . . . . . 10 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑑 ∈ 𝐴) → (𝐻‘𝑑) ∈ 𝐵)
7	6	ex 412	. . . . . . . . 9 ⊢ (𝐻:𝐴⟶𝐵 → (𝑑 ∈ 𝐴 → (𝐻‘𝑑) ∈ 𝐵))
8	5, 7	anim12d 609	. . . . . . . 8 ⊢ (𝐻:𝐴⟶𝐵 → ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) → ((𝐻‘𝑐) ∈ 𝐵 ∧ (𝐻‘𝑑) ∈ 𝐵)))
9	2, 3, 8	3syl 18	. . . . . . 7 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) → ((𝐻‘𝑐) ∈ 𝐵 ∧ (𝐻‘𝑑) ∈ 𝐵)))
10	9	imp 406	. . . . . 6 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) → ((𝐻‘𝑐) ∈ 𝐵 ∧ (𝐻‘𝑑) ∈ 𝐵))
11		breq1 5095	. . . . . . . 8 ⊢ (𝑎 = (𝐻‘𝑐) → (𝑎𝑆𝑏 ↔ (𝐻‘𝑐)𝑆𝑏))
12		eqeq1 2733	. . . . . . . 8 ⊢ (𝑎 = (𝐻‘𝑐) → (𝑎 = 𝑏 ↔ (𝐻‘𝑐) = 𝑏))
13		breq2 5096	. . . . . . . 8 ⊢ (𝑎 = (𝐻‘𝑐) → (𝑏𝑆𝑎 ↔ 𝑏𝑆(𝐻‘𝑐)))
14	11, 12, 13	3orbi123d 1437	. . . . . . 7 ⊢ (𝑎 = (𝐻‘𝑐) → ((𝑎𝑆𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑆𝑎) ↔ ((𝐻‘𝑐)𝑆𝑏 ∨ (𝐻‘𝑐) = 𝑏 ∨ 𝑏𝑆(𝐻‘𝑐))))
15		breq2 5096	. . . . . . . 8 ⊢ (𝑏 = (𝐻‘𝑑) → ((𝐻‘𝑐)𝑆𝑏 ↔ (𝐻‘𝑐)𝑆(𝐻‘𝑑)))
16		eqeq2 2741	. . . . . . . 8 ⊢ (𝑏 = (𝐻‘𝑑) → ((𝐻‘𝑐) = 𝑏 ↔ (𝐻‘𝑐) = (𝐻‘𝑑)))
17		breq1 5095	. . . . . . . 8 ⊢ (𝑏 = (𝐻‘𝑑) → (𝑏𝑆(𝐻‘𝑐) ↔ (𝐻‘𝑑)𝑆(𝐻‘𝑐)))
18	15, 16, 17	3orbi123d 1437	. . . . . . 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 3588	. . . . . 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 17	. . . . 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 7263	. . . . . 6 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) → (𝑐𝑅𝑑 ↔ (𝐻‘𝑐)𝑆(𝐻‘𝑑)))
22		f1of1 6763	. . . . . . . . 9 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴–1-1→𝐵)
23	2, 22	syl 17	. . . . . . . 8 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1→𝐵)
24		f1fveq 7199	. . . . . . . 8 ⊢ ((𝐻:𝐴–1-1→𝐵 ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) → ((𝐻‘𝑐) = (𝐻‘𝑑) ↔ 𝑐 = 𝑑))
25	23, 24	sylan 580	. . . . . . 7 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) → ((𝐻‘𝑐) = (𝐻‘𝑑) ↔ 𝑐 = 𝑑))
26	25	bicomd 223	. . . . . 6 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) → (𝑐 = 𝑑 ↔ (𝐻‘𝑐) = (𝐻‘𝑑)))
27		isorel 7263	. . . . . . 7 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) → (𝑑𝑅𝑐 ↔ (𝐻‘𝑑)𝑆(𝐻‘𝑐)))
28	27	ancom2s 650	. . . . . 6 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) → (𝑑𝑅𝑐 ↔ (𝐻‘𝑑)𝑆(𝐻‘𝑐)))
29	21, 26, 28	3orbi123d 1437	. . . . 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 259	. . . 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 3184	. . 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 609	. 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 5528	. 2 ⊢ (𝑆 Or 𝐵 ↔ (𝑆 Po 𝐵 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎𝑆𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑆𝑎)))
34		df-so 5528	. 2 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐴 (𝑐𝑅𝑑 ∨ 𝑐 = 𝑑 ∨ 𝑑𝑅𝑐)))
35	32, 33, 34	3imtr4g 296	1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Or 𝐵 → 𝑅 Or 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ w3o 1085   = wceq 1540   ∈ wcel 2109  ∀wral 3044   class class class wbr 5092   Po wpo 5525   Or wor 5526  ⟶wf 6478  –1-1→wf1 6479  –1-1-onto→wf1o 6481  ‘cfv 6482   Isom wiso 6483

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-id 5514  df-po 5527  df-so 5528  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-f1o 6489  df-fv 6490  df-isom 6491

This theorem is referenced by:  isoso  7285  isowe2  7287