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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 𝐵𝑅 Or 𝐴))

Proof of Theorem isosolem
Dummy variables 𝑎 𝑏 𝑐 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isopolem 5461 . . 3 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Po 𝐵𝑅 Po 𝐴))
2 df-3an 887 . . . . . . 7 ((𝑎𝐴𝑏𝐴𝑐𝐴) ↔ ((𝑎𝐴𝑏𝐴) ∧ 𝑐𝐴))
3 isof1o 5447 . . . . . . . . . . 11 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
4 f1of 5126 . . . . . . . . . . 11 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴𝐵)
5 ffvelrn 5300 . . . . . . . . . . . . 13 ((𝐻:𝐴𝐵𝑎𝐴) → (𝐻𝑎) ∈ 𝐵)
65ex 108 . . . . . . . . . . . 12 (𝐻:𝐴𝐵 → (𝑎𝐴 → (𝐻𝑎) ∈ 𝐵))
7 ffvelrn 5300 . . . . . . . . . . . . 13 ((𝐻:𝐴𝐵𝑏𝐴) → (𝐻𝑏) ∈ 𝐵)
87ex 108 . . . . . . . . . . . 12 (𝐻:𝐴𝐵 → (𝑏𝐴 → (𝐻𝑏) ∈ 𝐵))
9 ffvelrn 5300 . . . . . . . . . . . . 13 ((𝐻:𝐴𝐵𝑐𝐴) → (𝐻𝑐) ∈ 𝐵)
109ex 108 . . . . . . . . . . . 12 (𝐻:𝐴𝐵 → (𝑐𝐴 → (𝐻𝑐) ∈ 𝐵))
116, 8, 103anim123d 1214 . . . . . . . . . . 11 (𝐻:𝐴𝐵 → ((𝑎𝐴𝑏𝐴𝑐𝐴) → ((𝐻𝑎) ∈ 𝐵 ∧ (𝐻𝑏) ∈ 𝐵 ∧ (𝐻𝑐) ∈ 𝐵)))
123, 4, 113syl 17 . . . . . . . . . 10 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ((𝑎𝐴𝑏𝐴𝑐𝐴) → ((𝐻𝑎) ∈ 𝐵 ∧ (𝐻𝑏) ∈ 𝐵 ∧ (𝐻𝑐) ∈ 𝐵)))
1312imp 115 . . . . . . . . 9 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) → ((𝐻𝑎) ∈ 𝐵 ∧ (𝐻𝑏) ∈ 𝐵 ∧ (𝐻𝑐) ∈ 𝐵))
14 breq1 3767 . . . . . . . . . . 11 (𝑥 = (𝐻𝑎) → (𝑥𝑆𝑦 ↔ (𝐻𝑎)𝑆𝑦))
15 breq1 3767 . . . . . . . . . . . 12 (𝑥 = (𝐻𝑎) → (𝑥𝑆𝑧 ↔ (𝐻𝑎)𝑆𝑧))
1615orbi1d 705 . . . . . . . . . . 11 (𝑥 = (𝐻𝑎) → ((𝑥𝑆𝑧𝑧𝑆𝑦) ↔ ((𝐻𝑎)𝑆𝑧𝑧𝑆𝑦)))
1714, 16imbi12d 223 . . . . . . . . . 10 (𝑥 = (𝐻𝑎) → ((𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦)) ↔ ((𝐻𝑎)𝑆𝑦 → ((𝐻𝑎)𝑆𝑧𝑧𝑆𝑦))))
18 breq2 3768 . . . . . . . . . . 11 (𝑦 = (𝐻𝑏) → ((𝐻𝑎)𝑆𝑦 ↔ (𝐻𝑎)𝑆(𝐻𝑏)))
19 breq2 3768 . . . . . . . . . . . 12 (𝑦 = (𝐻𝑏) → (𝑧𝑆𝑦𝑧𝑆(𝐻𝑏)))
2019orbi2d 704 . . . . . . . . . . 11 (𝑦 = (𝐻𝑏) → (((𝐻𝑎)𝑆𝑧𝑧𝑆𝑦) ↔ ((𝐻𝑎)𝑆𝑧𝑧𝑆(𝐻𝑏))))
2118, 20imbi12d 223 . . . . . . . . . 10 (𝑦 = (𝐻𝑏) → (((𝐻𝑎)𝑆𝑦 → ((𝐻𝑎)𝑆𝑧𝑧𝑆𝑦)) ↔ ((𝐻𝑎)𝑆(𝐻𝑏) → ((𝐻𝑎)𝑆𝑧𝑧𝑆(𝐻𝑏)))))
22 breq2 3768 . . . . . . . . . . . 12 (𝑧 = (𝐻𝑐) → ((𝐻𝑎)𝑆𝑧 ↔ (𝐻𝑎)𝑆(𝐻𝑐)))
23 breq1 3767 . . . . . . . . . . . 12 (𝑧 = (𝐻𝑐) → (𝑧𝑆(𝐻𝑏) ↔ (𝐻𝑐)𝑆(𝐻𝑏)))
2422, 23orbi12d 707 . . . . . . . . . . 11 (𝑧 = (𝐻𝑐) → (((𝐻𝑎)𝑆𝑧𝑧𝑆(𝐻𝑏)) ↔ ((𝐻𝑎)𝑆(𝐻𝑐) ∨ (𝐻𝑐)𝑆(𝐻𝑏))))
2524imbi2d 219 . . . . . . . . . 10 (𝑧 = (𝐻𝑐) → (((𝐻𝑎)𝑆(𝐻𝑏) → ((𝐻𝑎)𝑆𝑧𝑧𝑆(𝐻𝑏))) ↔ ((𝐻𝑎)𝑆(𝐻𝑏) → ((𝐻𝑎)𝑆(𝐻𝑐) ∨ (𝐻𝑐)𝑆(𝐻𝑏)))))
2617, 21, 25rspc3v 2665 . . . . . . . . 9 (((𝐻𝑎) ∈ 𝐵 ∧ (𝐻𝑏) ∈ 𝐵 ∧ (𝐻𝑐) ∈ 𝐵) → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦)) → ((𝐻𝑎)𝑆(𝐻𝑏) → ((𝐻𝑎)𝑆(𝐻𝑐) ∨ (𝐻𝑐)𝑆(𝐻𝑏)))))
2713, 26syl 14 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦)) → ((𝐻𝑎)𝑆(𝐻𝑏) → ((𝐻𝑎)𝑆(𝐻𝑐) ∨ (𝐻𝑐)𝑆(𝐻𝑏)))))
28 isorel 5448 . . . . . . . . . 10 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎𝐴𝑏𝐴)) → (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)))
29283adantr3 1065 . . . . . . . . 9 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) → (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)))
30 isorel 5448 . . . . . . . . . . 11 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎𝐴𝑐𝐴)) → (𝑎𝑅𝑐 ↔ (𝐻𝑎)𝑆(𝐻𝑐)))
31303adantr2 1064 . . . . . . . . . 10 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) → (𝑎𝑅𝑐 ↔ (𝐻𝑎)𝑆(𝐻𝑐)))
32 isorel 5448 . . . . . . . . . . . 12 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐𝐴𝑏𝐴)) → (𝑐𝑅𝑏 ↔ (𝐻𝑐)𝑆(𝐻𝑏)))
3332ancom2s 500 . . . . . . . . . . 11 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑏𝐴𝑐𝐴)) → (𝑐𝑅𝑏 ↔ (𝐻𝑐)𝑆(𝐻𝑏)))
34333adantr1 1063 . . . . . . . . . 10 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) → (𝑐𝑅𝑏 ↔ (𝐻𝑐)𝑆(𝐻𝑏)))
3531, 34orbi12d 707 . . . . . . . . 9 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) → ((𝑎𝑅𝑐𝑐𝑅𝑏) ↔ ((𝐻𝑎)𝑆(𝐻𝑐) ∨ (𝐻𝑐)𝑆(𝐻𝑏))))
3629, 35imbi12d 223 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) → ((𝑎𝑅𝑏 → (𝑎𝑅𝑐𝑐𝑅𝑏)) ↔ ((𝐻𝑎)𝑆(𝐻𝑏) → ((𝐻𝑎)𝑆(𝐻𝑐) ∨ (𝐻𝑐)𝑆(𝐻𝑏)))))
3727, 36sylibrd 158 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦)) → (𝑎𝑅𝑏 → (𝑎𝑅𝑐𝑐𝑅𝑏))))
382, 37sylan2br 272 . . . . . 6 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ((𝑎𝐴𝑏𝐴) ∧ 𝑐𝐴)) → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦)) → (𝑎𝑅𝑏 → (𝑎𝑅𝑐𝑐𝑅𝑏))))
3938anassrs 380 . . . . 5 (((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎𝐴𝑏𝐴)) ∧ 𝑐𝐴) → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦)) → (𝑎𝑅𝑏 → (𝑎𝑅𝑐𝑐𝑅𝑏))))
4039ralrimdva 2399 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎𝐴𝑏𝐴)) → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦)) → ∀𝑐𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐𝑐𝑅𝑏))))
4140ralrimdvva 2404 . . 3 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦)) → ∀𝑎𝐴𝑏𝐴𝑐𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐𝑐𝑅𝑏))))
421, 41anim12d 318 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ((𝑆 Po 𝐵 ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦))) → (𝑅 Po 𝐴 ∧ ∀𝑎𝐴𝑏𝐴𝑐𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐𝑐𝑅𝑏)))))
43 df-iso 4034 . 2 (𝑆 Or 𝐵 ↔ (𝑆 Po 𝐵 ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦))))
44 df-iso 4034 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑎𝐴𝑏𝐴𝑐𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐𝑐𝑅𝑏))))
4542, 43, 443imtr4g 194 1 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Or 𝐵𝑅 Or 𝐴))
 Colors of variables: wff set class 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
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