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Theorem isosolem 5542
Description: Lemma for isoso 5543. (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 5540 . . 3 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Po 𝐵𝑅 Po 𝐴))
2 df-3an 922 . . . . . . 7 ((𝑎𝐴𝑏𝐴𝑐𝐴) ↔ ((𝑎𝐴𝑏𝐴) ∧ 𝑐𝐴))
3 isof1o 5526 . . . . . . . . . . 11 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
4 f1of 5201 . . . . . . . . . . 11 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴𝐵)
5 ffvelrn 5377 . . . . . . . . . . . . 13 ((𝐻:𝐴𝐵𝑎𝐴) → (𝐻𝑎) ∈ 𝐵)
65ex 113 . . . . . . . . . . . 12 (𝐻:𝐴𝐵 → (𝑎𝐴 → (𝐻𝑎) ∈ 𝐵))
7 ffvelrn 5377 . . . . . . . . . . . . 13 ((𝐻:𝐴𝐵𝑏𝐴) → (𝐻𝑏) ∈ 𝐵)
87ex 113 . . . . . . . . . . . 12 (𝐻:𝐴𝐵 → (𝑏𝐴 → (𝐻𝑏) ∈ 𝐵))
9 ffvelrn 5377 . . . . . . . . . . . . 13 ((𝐻:𝐴𝐵𝑐𝐴) → (𝐻𝑐) ∈ 𝐵)
109ex 113 . . . . . . . . . . . 12 (𝐻:𝐴𝐵 → (𝑐𝐴 → (𝐻𝑐) ∈ 𝐵))
116, 8, 103anim123d 1251 . . . . . . . . . . 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 122 . . . . . . . . 9 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) → ((𝐻𝑎) ∈ 𝐵 ∧ (𝐻𝑏) ∈ 𝐵 ∧ (𝐻𝑐) ∈ 𝐵))
14 breq1 3814 . . . . . . . . . . 11 (𝑥 = (𝐻𝑎) → (𝑥𝑆𝑦 ↔ (𝐻𝑎)𝑆𝑦))
15 breq1 3814 . . . . . . . . . . . 12 (𝑥 = (𝐻𝑎) → (𝑥𝑆𝑧 ↔ (𝐻𝑎)𝑆𝑧))
1615orbi1d 738 . . . . . . . . . . 11 (𝑥 = (𝐻𝑎) → ((𝑥𝑆𝑧𝑧𝑆𝑦) ↔ ((𝐻𝑎)𝑆𝑧𝑧𝑆𝑦)))
1714, 16imbi12d 232 . . . . . . . . . 10 (𝑥 = (𝐻𝑎) → ((𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦)) ↔ ((𝐻𝑎)𝑆𝑦 → ((𝐻𝑎)𝑆𝑧𝑧𝑆𝑦))))
18 breq2 3815 . . . . . . . . . . 11 (𝑦 = (𝐻𝑏) → ((𝐻𝑎)𝑆𝑦 ↔ (𝐻𝑎)𝑆(𝐻𝑏)))
19 breq2 3815 . . . . . . . . . . . 12 (𝑦 = (𝐻𝑏) → (𝑧𝑆𝑦𝑧𝑆(𝐻𝑏)))
2019orbi2d 737 . . . . . . . . . . 11 (𝑦 = (𝐻𝑏) → (((𝐻𝑎)𝑆𝑧𝑧𝑆𝑦) ↔ ((𝐻𝑎)𝑆𝑧𝑧𝑆(𝐻𝑏))))
2118, 20imbi12d 232 . . . . . . . . . 10 (𝑦 = (𝐻𝑏) → (((𝐻𝑎)𝑆𝑦 → ((𝐻𝑎)𝑆𝑧𝑧𝑆𝑦)) ↔ ((𝐻𝑎)𝑆(𝐻𝑏) → ((𝐻𝑎)𝑆𝑧𝑧𝑆(𝐻𝑏)))))
22 breq2 3815 . . . . . . . . . . . 12 (𝑧 = (𝐻𝑐) → ((𝐻𝑎)𝑆𝑧 ↔ (𝐻𝑎)𝑆(𝐻𝑐)))
23 breq1 3814 . . . . . . . . . . . 12 (𝑧 = (𝐻𝑐) → (𝑧𝑆(𝐻𝑏) ↔ (𝐻𝑐)𝑆(𝐻𝑏)))
2422, 23orbi12d 740 . . . . . . . . . . 11 (𝑧 = (𝐻𝑐) → (((𝐻𝑎)𝑆𝑧𝑧𝑆(𝐻𝑏)) ↔ ((𝐻𝑎)𝑆(𝐻𝑐) ∨ (𝐻𝑐)𝑆(𝐻𝑏))))
2524imbi2d 228 . . . . . . . . . 10 (𝑧 = (𝐻𝑐) → (((𝐻𝑎)𝑆(𝐻𝑏) → ((𝐻𝑎)𝑆𝑧𝑧𝑆(𝐻𝑏))) ↔ ((𝐻𝑎)𝑆(𝐻𝑏) → ((𝐻𝑎)𝑆(𝐻𝑐) ∨ (𝐻𝑐)𝑆(𝐻𝑏)))))
2617, 21, 25rspc3v 2726 . . . . . . . . 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 5527 . . . . . . . . . 10 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎𝐴𝑏𝐴)) → (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)))
29283adantr3 1100 . . . . . . . . 9 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) → (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)))
30 isorel 5527 . . . . . . . . . . 11 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎𝐴𝑐𝐴)) → (𝑎𝑅𝑐 ↔ (𝐻𝑎)𝑆(𝐻𝑐)))
31303adantr2 1099 . . . . . . . . . 10 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) → (𝑎𝑅𝑐 ↔ (𝐻𝑎)𝑆(𝐻𝑐)))
32 isorel 5527 . . . . . . . . . . . 12 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐𝐴𝑏𝐴)) → (𝑐𝑅𝑏 ↔ (𝐻𝑐)𝑆(𝐻𝑏)))
3332ancom2s 531 . . . . . . . . . . 11 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑏𝐴𝑐𝐴)) → (𝑐𝑅𝑏 ↔ (𝐻𝑐)𝑆(𝐻𝑏)))
34333adantr1 1098 . . . . . . . . . 10 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) → (𝑐𝑅𝑏 ↔ (𝐻𝑐)𝑆(𝐻𝑏)))
3531, 34orbi12d 740 . . . . . . . . 9 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) → ((𝑎𝑅𝑐𝑐𝑅𝑏) ↔ ((𝐻𝑎)𝑆(𝐻𝑐) ∨ (𝐻𝑐)𝑆(𝐻𝑏))))
3629, 35imbi12d 232 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) → ((𝑎𝑅𝑏 → (𝑎𝑅𝑐𝑐𝑅𝑏)) ↔ ((𝐻𝑎)𝑆(𝐻𝑏) → ((𝐻𝑎)𝑆(𝐻𝑐) ∨ (𝐻𝑐)𝑆(𝐻𝑏)))))
3727, 36sylibrd 167 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦)) → (𝑎𝑅𝑏 → (𝑎𝑅𝑐𝑐𝑅𝑏))))
382, 37sylan2br 282 . . . . . 6 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ((𝑎𝐴𝑏𝐴) ∧ 𝑐𝐴)) → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦)) → (𝑎𝑅𝑏 → (𝑎𝑅𝑐𝑐𝑅𝑏))))
3938anassrs 392 . . . . 5 (((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎𝐴𝑏𝐴)) ∧ 𝑐𝐴) → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦)) → (𝑎𝑅𝑏 → (𝑎𝑅𝑐𝑐𝑅𝑏))))
4039ralrimdva 2447 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎𝐴𝑏𝐴)) → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦)) → ∀𝑐𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐𝑐𝑅𝑏))))
4140ralrimdvva 2452 . . 3 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦)) → ∀𝑎𝐴𝑏𝐴𝑐𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐𝑐𝑅𝑏))))
421, 41anim12d 328 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ((𝑆 Po 𝐵 ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦))) → (𝑅 Po 𝐴 ∧ ∀𝑎𝐴𝑏𝐴𝑐𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐𝑐𝑅𝑏)))))
43 df-iso 4088 . 2 (𝑆 Or 𝐵 ↔ (𝑆 Po 𝐵 ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦))))
44 df-iso 4088 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑎𝐴𝑏𝐴𝑐𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐𝑐𝑅𝑏))))
4542, 43, 443imtr4g 203 1 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Or 𝐵𝑅 Or 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wo 662  w3a 920   = wceq 1285  wcel 1434  wral 2353   class class class wbr 3811   Po wpo 4085   Or wor 4086  wf 4965  1-1-ontowf1o 4968  cfv 4969   Isom wiso 4970
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-sbc 2827  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-id 4084  df-po 4087  df-iso 4088  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-f1o 4976  df-fv 4977  df-isom 4978
This theorem is referenced by:  isoso  5543
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