Theorem isosolem 5463

Description: Lemma for isoso 5464. (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 5461	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Po 𝐵 → 𝑅 Po 𝐴))
2		df-3an 887	. . . . . . 7 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ 𝐴))
3		isof1o 5447	. . . . . . . . . . 11 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
4		f1of 5126	. . . . . . . . . . 11 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴⟶𝐵)
5		ffvelrn 5300	. . . . . . . . . . . . 13 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑎 ∈ 𝐴) → (𝐻‘𝑎) ∈ 𝐵)
6	5	ex 108	. . . . . . . . . . . 12 ⊢ (𝐻:𝐴⟶𝐵 → (𝑎 ∈ 𝐴 → (𝐻‘𝑎) ∈ 𝐵))
7		ffvelrn 5300	. . . . . . . . . . . . 13 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑏 ∈ 𝐴) → (𝐻‘𝑏) ∈ 𝐵)
8	7	ex 108	. . . . . . . . . . . 12 ⊢ (𝐻:𝐴⟶𝐵 → (𝑏 ∈ 𝐴 → (𝐻‘𝑏) ∈ 𝐵))
9		ffvelrn 5300	. . . . . . . . . . . . 13 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑐 ∈ 𝐴) → (𝐻‘𝑐) ∈ 𝐵)
10	9	ex 108	. . . . . . . . . . . 12 ⊢ (𝐻:𝐴⟶𝐵 → (𝑐 ∈ 𝐴 → (𝐻‘𝑐) ∈ 𝐵))
11	6, 8, 10	3anim123d 1214	. . . . . . . . . . 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 115	. . . . . . . . 9 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) → ((𝐻‘𝑎) ∈ 𝐵 ∧ (𝐻‘𝑏) ∈ 𝐵 ∧ (𝐻‘𝑐) ∈ 𝐵))
14		breq1 3767	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐻‘𝑎) → (𝑥𝑆𝑦 ↔ (𝐻‘𝑎)𝑆𝑦))
15		breq1 3767	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐻‘𝑎) → (𝑥𝑆𝑧 ↔ (𝐻‘𝑎)𝑆𝑧))
16	15	orbi1d 705	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐻‘𝑎) → ((𝑥𝑆𝑧 ∨ 𝑧𝑆𝑦) ↔ ((𝐻‘𝑎)𝑆𝑧 ∨ 𝑧𝑆𝑦)))
17	14, 16	imbi12d 223	. . . . . . . . . 10 ⊢ (𝑥 = (𝐻‘𝑎) → ((𝑥𝑆𝑦 → (𝑥𝑆𝑧 ∨ 𝑧𝑆𝑦)) ↔ ((𝐻‘𝑎)𝑆𝑦 → ((𝐻‘𝑎)𝑆𝑧 ∨ 𝑧𝑆𝑦))))
18		breq2 3768	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐻‘𝑏) → ((𝐻‘𝑎)𝑆𝑦 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏)))
19		breq2 3768	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐻‘𝑏) → (𝑧𝑆𝑦 ↔ 𝑧𝑆(𝐻‘𝑏)))
20	19	orbi2d 704	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐻‘𝑏) → (((𝐻‘𝑎)𝑆𝑧 ∨ 𝑧𝑆𝑦) ↔ ((𝐻‘𝑎)𝑆𝑧 ∨ 𝑧𝑆(𝐻‘𝑏))))
21	18, 20	imbi12d 223	. . . . . . . . . 10 ⊢ (𝑦 = (𝐻‘𝑏) → (((𝐻‘𝑎)𝑆𝑦 → ((𝐻‘𝑎)𝑆𝑧 ∨ 𝑧𝑆𝑦)) ↔ ((𝐻‘𝑎)𝑆(𝐻‘𝑏) → ((𝐻‘𝑎)𝑆𝑧 ∨ 𝑧𝑆(𝐻‘𝑏)))))
22		breq2 3768	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐻‘𝑐) → ((𝐻‘𝑎)𝑆𝑧 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑐)))
23		breq1 3767	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐻‘𝑐) → (𝑧𝑆(𝐻‘𝑏) ↔ (𝐻‘𝑐)𝑆(𝐻‘𝑏)))
24	22, 23	orbi12d 707	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐻‘𝑐) → (((𝐻‘𝑎)𝑆𝑧 ∨ 𝑧𝑆(𝐻‘𝑏)) ↔ ((𝐻‘𝑎)𝑆(𝐻‘𝑐) ∨ (𝐻‘𝑐)𝑆(𝐻‘𝑏))))
25	24	imbi2d 219	. . . . . . . . . 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 2665	. . . . . . . . 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 5448	. . . . . . . . . 10 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (𝑎𝑅𝑏 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏)))
29	28	3adantr3 1065	. . . . . . . . 9 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) → (𝑎𝑅𝑏 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏)))
30		isorel 5448	. . . . . . . . . . 11 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) → (𝑎𝑅𝑐 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑐)))
31	30	3adantr2 1064	. . . . . . . . . 10 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) → (𝑎𝑅𝑐 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑐)))
32		isorel 5448	. . . . . . . . . . . 12 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (𝑐𝑅𝑏 ↔ (𝐻‘𝑐)𝑆(𝐻‘𝑏)))
33	32	ancom2s 500	. . . . . . . . . . 11 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) → (𝑐𝑅𝑏 ↔ (𝐻‘𝑐)𝑆(𝐻‘𝑏)))
34	33	3adantr1 1063	. . . . . . . . . 10 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) → (𝑐𝑅𝑏 ↔ (𝐻‘𝑐)𝑆(𝐻‘𝑏)))
35	31, 34	orbi12d 707	. . . . . . . . 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 223	. . . . . . . 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 158	. . . . . . 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 272	. . . . . 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 380	. . . . 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 2399	. . . 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 2404	. . 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 318	. 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 4034	. 2 ⊢ (𝑆 Or 𝐵 ↔ (𝑆 Po 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧 ∨ 𝑧𝑆𝑦))))
44		df-iso 4034	. 2 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏))))
45	42, 43, 44	3imtr4g 194	1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Or 𝐵 → 𝑅 Or 𝐴))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629   ∧ w3a 885   = wceq 1243   ∈ wcel 1393  ∀wral 2306   class class class wbr 3764   Po wpo 4031   Or wor 4032  ⟶wf 4898  –1-1-onto→wf1o 4901  ‘cfv 4902   Isom wiso 4903

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-po 4033  df-iso 4034  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-f1o 4909  df-fv 4910  df-isom 4911

This theorem is referenced by:  isoso  5464