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Theorem isosolem 7368
Description: Lemma for isoso 7369. (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 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isopolem 7366 . . 3 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Po 𝐵𝑅 Po 𝐴))
2 isof1o 7344 . . . . . . . 8 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
3 f1of 6847 . . . . . . . 8 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴𝐵)
4 ffvelcdm 7100 . . . . . . . . . 10 ((𝐻:𝐴𝐵𝑐𝐴) → (𝐻𝑐) ∈ 𝐵)
54ex 412 . . . . . . . . 9 (𝐻:𝐴𝐵 → (𝑐𝐴 → (𝐻𝑐) ∈ 𝐵))
6 ffvelcdm 7100 . . . . . . . . . 10 ((𝐻:𝐴𝐵𝑑𝐴) → (𝐻𝑑) ∈ 𝐵)
76ex 412 . . . . . . . . 9 (𝐻:𝐴𝐵 → (𝑑𝐴 → (𝐻𝑑) ∈ 𝐵))
85, 7anim12d 609 . . . . . . . 8 (𝐻:𝐴𝐵 → ((𝑐𝐴𝑑𝐴) → ((𝐻𝑐) ∈ 𝐵 ∧ (𝐻𝑑) ∈ 𝐵)))
92, 3, 83syl 18 . . . . . . 7 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ((𝑐𝐴𝑑𝐴) → ((𝐻𝑐) ∈ 𝐵 ∧ (𝐻𝑑) ∈ 𝐵)))
109imp 406 . . . . . 6 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐𝐴𝑑𝐴)) → ((𝐻𝑐) ∈ 𝐵 ∧ (𝐻𝑑) ∈ 𝐵))
11 breq1 5145 . . . . . . . 8 (𝑎 = (𝐻𝑐) → (𝑎𝑆𝑏 ↔ (𝐻𝑐)𝑆𝑏))
12 eqeq1 2740 . . . . . . . 8 (𝑎 = (𝐻𝑐) → (𝑎 = 𝑏 ↔ (𝐻𝑐) = 𝑏))
13 breq2 5146 . . . . . . . 8 (𝑎 = (𝐻𝑐) → (𝑏𝑆𝑎𝑏𝑆(𝐻𝑐)))
1411, 12, 133orbi123d 1436 . . . . . . 7 (𝑎 = (𝐻𝑐) → ((𝑎𝑆𝑏𝑎 = 𝑏𝑏𝑆𝑎) ↔ ((𝐻𝑐)𝑆𝑏 ∨ (𝐻𝑐) = 𝑏𝑏𝑆(𝐻𝑐))))
15 breq2 5146 . . . . . . . 8 (𝑏 = (𝐻𝑑) → ((𝐻𝑐)𝑆𝑏 ↔ (𝐻𝑐)𝑆(𝐻𝑑)))
16 eqeq2 2748 . . . . . . . 8 (𝑏 = (𝐻𝑑) → ((𝐻𝑐) = 𝑏 ↔ (𝐻𝑐) = (𝐻𝑑)))
17 breq1 5145 . . . . . . . 8 (𝑏 = (𝐻𝑑) → (𝑏𝑆(𝐻𝑐) ↔ (𝐻𝑑)𝑆(𝐻𝑐)))
1815, 16, 173orbi123d 1436 . . . . . . 7 (𝑏 = (𝐻𝑑) → (((𝐻𝑐)𝑆𝑏 ∨ (𝐻𝑐) = 𝑏𝑏𝑆(𝐻𝑐)) ↔ ((𝐻𝑐)𝑆(𝐻𝑑) ∨ (𝐻𝑐) = (𝐻𝑑) ∨ (𝐻𝑑)𝑆(𝐻𝑐))))
1914, 18rspc2v 3632 . . . . . 6 (((𝐻𝑐) ∈ 𝐵 ∧ (𝐻𝑑) ∈ 𝐵) → (∀𝑎𝐵𝑏𝐵 (𝑎𝑆𝑏𝑎 = 𝑏𝑏𝑆𝑎) → ((𝐻𝑐)𝑆(𝐻𝑑) ∨ (𝐻𝑐) = (𝐻𝑑) ∨ (𝐻𝑑)𝑆(𝐻𝑐))))
2010, 19syl 17 . . . . 5 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐𝐴𝑑𝐴)) → (∀𝑎𝐵𝑏𝐵 (𝑎𝑆𝑏𝑎 = 𝑏𝑏𝑆𝑎) → ((𝐻𝑐)𝑆(𝐻𝑑) ∨ (𝐻𝑐) = (𝐻𝑑) ∨ (𝐻𝑑)𝑆(𝐻𝑐))))
21 isorel 7347 . . . . . 6 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐𝐴𝑑𝐴)) → (𝑐𝑅𝑑 ↔ (𝐻𝑐)𝑆(𝐻𝑑)))
22 f1of1 6846 . . . . . . . . 9 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴1-1𝐵)
232, 22syl 17 . . . . . . . 8 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1𝐵)
24 f1fveq 7283 . . . . . . . 8 ((𝐻:𝐴1-1𝐵 ∧ (𝑐𝐴𝑑𝐴)) → ((𝐻𝑐) = (𝐻𝑑) ↔ 𝑐 = 𝑑))
2523, 24sylan 580 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐𝐴𝑑𝐴)) → ((𝐻𝑐) = (𝐻𝑑) ↔ 𝑐 = 𝑑))
2625bicomd 223 . . . . . 6 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐𝐴𝑑𝐴)) → (𝑐 = 𝑑 ↔ (𝐻𝑐) = (𝐻𝑑)))
27 isorel 7347 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑐𝐴)) → (𝑑𝑅𝑐 ↔ (𝐻𝑑)𝑆(𝐻𝑐)))
2827ancom2s 650 . . . . . 6 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐𝐴𝑑𝐴)) → (𝑑𝑅𝑐 ↔ (𝐻𝑑)𝑆(𝐻𝑐)))
2921, 26, 283orbi123d 1436 . . . . 5 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐𝐴𝑑𝐴)) → ((𝑐𝑅𝑑𝑐 = 𝑑𝑑𝑅𝑐) ↔ ((𝐻𝑐)𝑆(𝐻𝑑) ∨ (𝐻𝑐) = (𝐻𝑑) ∨ (𝐻𝑑)𝑆(𝐻𝑐))))
3020, 29sylibrd 259 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐𝐴𝑑𝐴)) → (∀𝑎𝐵𝑏𝐵 (𝑎𝑆𝑏𝑎 = 𝑏𝑏𝑆𝑎) → (𝑐𝑅𝑑𝑐 = 𝑑𝑑𝑅𝑐)))
3130ralrimdvva 3210 . . 3 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑎𝐵𝑏𝐵 (𝑎𝑆𝑏𝑎 = 𝑏𝑏𝑆𝑎) → ∀𝑐𝐴𝑑𝐴 (𝑐𝑅𝑑𝑐 = 𝑑𝑑𝑅𝑐)))
321, 31anim12d 609 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ((𝑆 Po 𝐵 ∧ ∀𝑎𝐵𝑏𝐵 (𝑎𝑆𝑏𝑎 = 𝑏𝑏𝑆𝑎)) → (𝑅 Po 𝐴 ∧ ∀𝑐𝐴𝑑𝐴 (𝑐𝑅𝑑𝑐 = 𝑑𝑑𝑅𝑐))))
33 df-so 5592 . 2 (𝑆 Or 𝐵 ↔ (𝑆 Po 𝐵 ∧ ∀𝑎𝐵𝑏𝐵 (𝑎𝑆𝑏𝑎 = 𝑏𝑏𝑆𝑎)))
34 df-so 5592 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑐𝐴𝑑𝐴 (𝑐𝑅𝑑𝑐 = 𝑑𝑑𝑅𝑐)))
3532, 33, 343imtr4g 296 1 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Or 𝐵𝑅 Or 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3o 1085   = wceq 1539  wcel 2107  wral 3060   class class class wbr 5142   Po wpo 5589   Or wor 5590  wf 6556  1-1wf1 6557  1-1-ontowf1o 6559  cfv 6560   Isom wiso 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-po 5591  df-so 5592  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-f1o 6567  df-fv 6568  df-isom 6569
This theorem is referenced by:  isoso  7369  isowe2  7371
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