Theorem isosolem 5483

Description: Lemma for isoso 5484. (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 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isopolem 5481	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Po 𝐵 → 𝑅 Po 𝐴))
2		df-3an 921	. . . . . . 7 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ 𝐴))
3		isof1o 5467	. . . . . . . . . . 11 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
4		f1of 5146	. . . . . . . . . . 11 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴⟶𝐵)
5		ffvelrn 5321	. . . . . . . . . . . . 13 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑎 ∈ 𝐴) → (𝐻‘𝑎) ∈ 𝐵)
6	5	ex 113	. . . . . . . . . . . 12 ⊢ (𝐻:𝐴⟶𝐵 → (𝑎 ∈ 𝐴 → (𝐻‘𝑎) ∈ 𝐵))
7		ffvelrn 5321	. . . . . . . . . . . . 13 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑏 ∈ 𝐴) → (𝐻‘𝑏) ∈ 𝐵)
8	7	ex 113	. . . . . . . . . . . 12 ⊢ (𝐻:𝐴⟶𝐵 → (𝑏 ∈ 𝐴 → (𝐻‘𝑏) ∈ 𝐵))
9		ffvelrn 5321	. . . . . . . . . . . . 13 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑐 ∈ 𝐴) → (𝐻‘𝑐) ∈ 𝐵)
10	9	ex 113	. . . . . . . . . . . 12 ⊢ (𝐻:𝐴⟶𝐵 → (𝑐 ∈ 𝐴 → (𝐻‘𝑐) ∈ 𝐵))
11	6, 8, 10	3anim123d 1250	. . . . . . . . . . 11 ⊢ (𝐻:𝐴⟶𝐵 → ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → ((𝐻‘𝑎) ∈ 𝐵 ∧ (𝐻‘𝑏) ∈ 𝐵 ∧ (𝐻‘𝑐) ∈ 𝐵)))
12	3, 4, 11	3syl 17	. . . . . . . . . 10 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → ((𝐻‘𝑎) ∈ 𝐵 ∧ (𝐻‘𝑏) ∈ 𝐵 ∧ (𝐻‘𝑐) ∈ 𝐵)))
13	12	imp 122	. . . . . . . . 9 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) → ((𝐻‘𝑎) ∈ 𝐵 ∧ (𝐻‘𝑏) ∈ 𝐵 ∧ (𝐻‘𝑐) ∈ 𝐵))
14		breq1 3788	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐻‘𝑎) → (𝑥𝑆𝑦 ↔ (𝐻‘𝑎)𝑆𝑦))
15		breq1 3788	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐻‘𝑎) → (𝑥𝑆𝑧 ↔ (𝐻‘𝑎)𝑆𝑧))
16	15	orbi1d 737	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐻‘𝑎) → ((𝑥𝑆𝑧 ∨ 𝑧𝑆𝑦) ↔ ((𝐻‘𝑎)𝑆𝑧 ∨ 𝑧𝑆𝑦)))
17	14, 16	imbi12d 232	. . . . . . . . . 10 ⊢ (𝑥 = (𝐻‘𝑎) → ((𝑥𝑆𝑦 → (𝑥𝑆𝑧 ∨ 𝑧𝑆𝑦)) ↔ ((𝐻‘𝑎)𝑆𝑦 → ((𝐻‘𝑎)𝑆𝑧 ∨ 𝑧𝑆𝑦))))
18		breq2 3789	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐻‘𝑏) → ((𝐻‘𝑎)𝑆𝑦 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏)))
19		breq2 3789	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐻‘𝑏) → (𝑧𝑆𝑦 ↔ 𝑧𝑆(𝐻‘𝑏)))
20	19	orbi2d 736	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐻‘𝑏) → (((𝐻‘𝑎)𝑆𝑧 ∨ 𝑧𝑆𝑦) ↔ ((𝐻‘𝑎)𝑆𝑧 ∨ 𝑧𝑆(𝐻‘𝑏))))
21	18, 20	imbi12d 232	. . . . . . . . . 10 ⊢ (𝑦 = (𝐻‘𝑏) → (((𝐻‘𝑎)𝑆𝑦 → ((𝐻‘𝑎)𝑆𝑧 ∨ 𝑧𝑆𝑦)) ↔ ((𝐻‘𝑎)𝑆(𝐻‘𝑏) → ((𝐻‘𝑎)𝑆𝑧 ∨ 𝑧𝑆(𝐻‘𝑏)))))
22		breq2 3789	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐻‘𝑐) → ((𝐻‘𝑎)𝑆𝑧 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑐)))
23		breq1 3788	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐻‘𝑐) → (𝑧𝑆(𝐻‘𝑏) ↔ (𝐻‘𝑐)𝑆(𝐻‘𝑏)))
24	22, 23	orbi12d 739	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐻‘𝑐) → (((𝐻‘𝑎)𝑆𝑧 ∨ 𝑧𝑆(𝐻‘𝑏)) ↔ ((𝐻‘𝑎)𝑆(𝐻‘𝑐) ∨ (𝐻‘𝑐)𝑆(𝐻‘𝑏))))
25	24	imbi2d 228	. . . . . . . . . 10 ⊢ (𝑧 = (𝐻‘𝑐) → (((𝐻‘𝑎)𝑆(𝐻‘𝑏) → ((𝐻‘𝑎)𝑆𝑧 ∨ 𝑧𝑆(𝐻‘𝑏))) ↔ ((𝐻‘𝑎)𝑆(𝐻‘𝑏) → ((𝐻‘𝑎)𝑆(𝐻‘𝑐) ∨ (𝐻‘𝑐)𝑆(𝐻‘𝑏)))))
26	17, 21, 25	rspc3v 2716	. . . . . . . . 9 ⊢ (((𝐻‘𝑎) ∈ 𝐵 ∧ (𝐻‘𝑏) ∈ 𝐵 ∧ (𝐻‘𝑐) ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧 ∨ 𝑧𝑆𝑦)) → ((𝐻‘𝑎)𝑆(𝐻‘𝑏) → ((𝐻‘𝑎)𝑆(𝐻‘𝑐) ∨ (𝐻‘𝑐)𝑆(𝐻‘𝑏)))))
27	13, 26	syl 14	. . . . . . . 8 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧 ∨ 𝑧𝑆𝑦)) → ((𝐻‘𝑎)𝑆(𝐻‘𝑏) → ((𝐻‘𝑎)𝑆(𝐻‘𝑐) ∨ (𝐻‘𝑐)𝑆(𝐻‘𝑏)))))
28		isorel 5468	. . . . . . . . . 10 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (𝑎𝑅𝑏 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏)))
29	28	3adantr3 1099	. . . . . . . . 9 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) → (𝑎𝑅𝑏 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏)))
30		isorel 5468	. . . . . . . . . . 11 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) → (𝑎𝑅𝑐 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑐)))
31	30	3adantr2 1098	. . . . . . . . . 10 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) → (𝑎𝑅𝑐 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑐)))
32		isorel 5468	. . . . . . . . . . . 12 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (𝑐𝑅𝑏 ↔ (𝐻‘𝑐)𝑆(𝐻‘𝑏)))
33	32	ancom2s 530	. . . . . . . . . . 11 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) → (𝑐𝑅𝑏 ↔ (𝐻‘𝑐)𝑆(𝐻‘𝑏)))
34	33	3adantr1 1097	. . . . . . . . . 10 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) → (𝑐𝑅𝑏 ↔ (𝐻‘𝑐)𝑆(𝐻‘𝑏)))
35	31, 34	orbi12d 739	. . . . . . . . 9 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) → ((𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏) ↔ ((𝐻‘𝑎)𝑆(𝐻‘𝑐) ∨ (𝐻‘𝑐)𝑆(𝐻‘𝑏))))
36	29, 35	imbi12d 232	. . . . . . . 8 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) → ((𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏)) ↔ ((𝐻‘𝑎)𝑆(𝐻‘𝑏) → ((𝐻‘𝑎)𝑆(𝐻‘𝑐) ∨ (𝐻‘𝑐)𝑆(𝐻‘𝑏)))))
37	27, 36	sylibrd 167	. . . . . . 7 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧 ∨ 𝑧𝑆𝑦)) → (𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏))))
38	2, 37	sylan2br 282	. . . . . 6 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ 𝐴)) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧 ∨ 𝑧𝑆𝑦)) → (𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏))))
39	38	anassrs 392	. . . . 5 ⊢ (((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) ∧ 𝑐 ∈ 𝐴) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧 ∨ 𝑧𝑆𝑦)) → (𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏))))
40	39	ralrimdva 2441	. . . 4 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧 ∨ 𝑧𝑆𝑦)) → ∀𝑐 ∈ 𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏))))
41	40	ralrimdvva 2446	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧 ∨ 𝑧𝑆𝑦)) → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏))))
42	1, 41	anim12d 328	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ((𝑆 Po 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧 ∨ 𝑧𝑆𝑦))) → (𝑅 Po 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏)))))
43		df-iso 4052	. 2 ⊢ (𝑆 Or 𝐵 ↔ (𝑆 Po 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧 ∨ 𝑧𝑆𝑦))))
44		df-iso 4052	. 2 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏))))
45	42, 43, 44	3imtr4g 203	1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Or 𝐵 → 𝑅 Or 𝐴))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   ∨ wo 661   ∧ w3a 919   = wceq 1284   ∈ wcel 1433  ∀wral 2348   class class class wbr 3785   Po wpo 4049   Or wor 4050  ⟶wf 4918  –1-1-onto→wf1o 4921  ‘cfv 4922   Isom wiso 4923

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-po 4051  df-iso 4052  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-f1o 4929  df-fv 4930  df-isom 4931

This theorem is referenced by:  isoso  5484