Theorem isosolem 6497

Description: Lemma for isoso 6498. (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 6495	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Po 𝐵 → 𝑅 Po 𝐴))
2		isof1o 6473	. . . . . . . 8 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
3		f1of 6050	. . . . . . . 8 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴⟶𝐵)
4		ffvelrn 6265	. . . . . . . . . 10 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑐 ∈ 𝐴) → (𝐻‘𝑐) ∈ 𝐵)
5	4	ex 449	. . . . . . . . 9 ⊢ (𝐻:𝐴⟶𝐵 → (𝑐 ∈ 𝐴 → (𝐻‘𝑐) ∈ 𝐵))
6		ffvelrn 6265	. . . . . . . . . 10 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑑 ∈ 𝐴) → (𝐻‘𝑑) ∈ 𝐵)
7	6	ex 449	. . . . . . . . 9 ⊢ (𝐻:𝐴⟶𝐵 → (𝑑 ∈ 𝐴 → (𝐻‘𝑑) ∈ 𝐵))
8	5, 7	anim12d 584	. . . . . . . 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 444	. . . . . 6 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) → ((𝐻‘𝑐) ∈ 𝐵 ∧ (𝐻‘𝑑) ∈ 𝐵))
11		breq1 4586	. . . . . . . 8 ⊢ (𝑎 = (𝐻‘𝑐) → (𝑎𝑆𝑏 ↔ (𝐻‘𝑐)𝑆𝑏))
12		eqeq1 2614	. . . . . . . 8 ⊢ (𝑎 = (𝐻‘𝑐) → (𝑎 = 𝑏 ↔ (𝐻‘𝑐) = 𝑏))
13		breq2 4587	. . . . . . . 8 ⊢ (𝑎 = (𝐻‘𝑐) → (𝑏𝑆𝑎 ↔ 𝑏𝑆(𝐻‘𝑐)))
14	11, 12, 13	3orbi123d 1390	. . . . . . 7 ⊢ (𝑎 = (𝐻‘𝑐) → ((𝑎𝑆𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑆𝑎) ↔ ((𝐻‘𝑐)𝑆𝑏 ∨ (𝐻‘𝑐) = 𝑏 ∨ 𝑏𝑆(𝐻‘𝑐))))
15		breq2 4587	. . . . . . . 8 ⊢ (𝑏 = (𝐻‘𝑑) → ((𝐻‘𝑐)𝑆𝑏 ↔ (𝐻‘𝑐)𝑆(𝐻‘𝑑)))
16		eqeq2 2621	. . . . . . . 8 ⊢ (𝑏 = (𝐻‘𝑑) → ((𝐻‘𝑐) = 𝑏 ↔ (𝐻‘𝑐) = (𝐻‘𝑑)))
17		breq1 4586	. . . . . . . 8 ⊢ (𝑏 = (𝐻‘𝑑) → (𝑏𝑆(𝐻‘𝑐) ↔ (𝐻‘𝑑)𝑆(𝐻‘𝑐)))
18	15, 16, 17	3orbi123d 1390	. . . . . . 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 3293	. . . . . 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 6476	. . . . . 6 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) → (𝑐𝑅𝑑 ↔ (𝐻‘𝑐)𝑆(𝐻‘𝑑)))
22		f1of1 6049	. . . . . . . . 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 6420	. . . . . . . 8 ⊢ ((𝐻:𝐴–1-1→𝐵 ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) → ((𝐻‘𝑐) = (𝐻‘𝑑) ↔ 𝑐 = 𝑑))
25	23, 24	sylan 487	. . . . . . 7 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) → ((𝐻‘𝑐) = (𝐻‘𝑑) ↔ 𝑐 = 𝑑))
26	25	bicomd 212	. . . . . 6 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) → (𝑐 = 𝑑 ↔ (𝐻‘𝑐) = (𝐻‘𝑑)))
27		isorel 6476	. . . . . . 7 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) → (𝑑𝑅𝑐 ↔ (𝐻‘𝑑)𝑆(𝐻‘𝑐)))
28	27	ancom2s 840	. . . . . 6 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) → (𝑑𝑅𝑐 ↔ (𝐻‘𝑑)𝑆(𝐻‘𝑐)))
29	21, 26, 28	3orbi123d 1390	. . . . 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 248	. . . 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 2957	. . 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 584	. 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 4960	. 2 ⊢ (𝑆 Or 𝐵 ↔ (𝑆 Po 𝐵 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎𝑆𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑆𝑎)))
34		df-so 4960	. 2 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐴 (𝑐𝑅𝑑 ∨ 𝑐 = 𝑑 ∨ 𝑑𝑅𝑐)))
35	32, 33, 34	3imtr4g 284	1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Or 𝐵 → 𝑅 Or 𝐴))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∨ w3o 1030   = wceq 1475   ∈ wcel 1977  ∀wral 2896   class class class wbr 4583   Po wpo 4957   Or wor 4958  ⟶wf 5800  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804   Isom wiso 5805

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-f1o 5811  df-fv 5812  df-isom 5813

This theorem is referenced by:  isoso  6498  isowe2  6500