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Theorem isosolem 6557
Description: Lemma for isoso 6558. (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 6555 . . 3 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Po 𝐵𝑅 Po 𝐴))
2 isof1o 6533 . . . . . . . 8 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
3 f1of 6099 . . . . . . . 8 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴𝐵)
4 ffvelrn 6318 . . . . . . . . . 10 ((𝐻:𝐴𝐵𝑐𝐴) → (𝐻𝑐) ∈ 𝐵)
54ex 450 . . . . . . . . 9 (𝐻:𝐴𝐵 → (𝑐𝐴 → (𝐻𝑐) ∈ 𝐵))
6 ffvelrn 6318 . . . . . . . . . 10 ((𝐻:𝐴𝐵𝑑𝐴) → (𝐻𝑑) ∈ 𝐵)
76ex 450 . . . . . . . . 9 (𝐻:𝐴𝐵 → (𝑑𝐴 → (𝐻𝑑) ∈ 𝐵))
85, 7anim12d 585 . . . . . . . 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 445 . . . . . 6 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐𝐴𝑑𝐴)) → ((𝐻𝑐) ∈ 𝐵 ∧ (𝐻𝑑) ∈ 𝐵))
11 breq1 4621 . . . . . . . 8 (𝑎 = (𝐻𝑐) → (𝑎𝑆𝑏 ↔ (𝐻𝑐)𝑆𝑏))
12 eqeq1 2625 . . . . . . . 8 (𝑎 = (𝐻𝑐) → (𝑎 = 𝑏 ↔ (𝐻𝑐) = 𝑏))
13 breq2 4622 . . . . . . . 8 (𝑎 = (𝐻𝑐) → (𝑏𝑆𝑎𝑏𝑆(𝐻𝑐)))
1411, 12, 133orbi123d 1395 . . . . . . 7 (𝑎 = (𝐻𝑐) → ((𝑎𝑆𝑏𝑎 = 𝑏𝑏𝑆𝑎) ↔ ((𝐻𝑐)𝑆𝑏 ∨ (𝐻𝑐) = 𝑏𝑏𝑆(𝐻𝑐))))
15 breq2 4622 . . . . . . . 8 (𝑏 = (𝐻𝑑) → ((𝐻𝑐)𝑆𝑏 ↔ (𝐻𝑐)𝑆(𝐻𝑑)))
16 eqeq2 2632 . . . . . . . 8 (𝑏 = (𝐻𝑑) → ((𝐻𝑐) = 𝑏 ↔ (𝐻𝑐) = (𝐻𝑑)))
17 breq1 4621 . . . . . . . 8 (𝑏 = (𝐻𝑑) → (𝑏𝑆(𝐻𝑐) ↔ (𝐻𝑑)𝑆(𝐻𝑐)))
1815, 16, 173orbi123d 1395 . . . . . . 7 (𝑏 = (𝐻𝑑) → (((𝐻𝑐)𝑆𝑏 ∨ (𝐻𝑐) = 𝑏𝑏𝑆(𝐻𝑐)) ↔ ((𝐻𝑐)𝑆(𝐻𝑑) ∨ (𝐻𝑐) = (𝐻𝑑) ∨ (𝐻𝑑)𝑆(𝐻𝑐))))
1914, 18rspc2v 3310 . . . . . 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 6536 . . . . . 6 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐𝐴𝑑𝐴)) → (𝑐𝑅𝑑 ↔ (𝐻𝑐)𝑆(𝐻𝑑)))
22 f1of1 6098 . . . . . . . . 9 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴1-1𝐵)
232, 22syl 17 . . . . . . . 8 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1𝐵)
24 f1fveq 6479 . . . . . . . 8 ((𝐻:𝐴1-1𝐵 ∧ (𝑐𝐴𝑑𝐴)) → ((𝐻𝑐) = (𝐻𝑑) ↔ 𝑐 = 𝑑))
2523, 24sylan 488 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐𝐴𝑑𝐴)) → ((𝐻𝑐) = (𝐻𝑑) ↔ 𝑐 = 𝑑))
2625bicomd 213 . . . . . 6 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐𝐴𝑑𝐴)) → (𝑐 = 𝑑 ↔ (𝐻𝑐) = (𝐻𝑑)))
27 isorel 6536 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑐𝐴)) → (𝑑𝑅𝑐 ↔ (𝐻𝑑)𝑆(𝐻𝑐)))
2827ancom2s 843 . . . . . 6 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐𝐴𝑑𝐴)) → (𝑑𝑅𝑐 ↔ (𝐻𝑑)𝑆(𝐻𝑐)))
2921, 26, 283orbi123d 1395 . . . . 5 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐𝐴𝑑𝐴)) → ((𝑐𝑅𝑑𝑐 = 𝑑𝑑𝑅𝑐) ↔ ((𝐻𝑐)𝑆(𝐻𝑑) ∨ (𝐻𝑐) = (𝐻𝑑) ∨ (𝐻𝑑)𝑆(𝐻𝑐))))
3020, 29sylibrd 249 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐𝐴𝑑𝐴)) → (∀𝑎𝐵𝑏𝐵 (𝑎𝑆𝑏𝑎 = 𝑏𝑏𝑆𝑎) → (𝑐𝑅𝑑𝑐 = 𝑑𝑑𝑅𝑐)))
3130ralrimdvva 2969 . . 3 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑎𝐵𝑏𝐵 (𝑎𝑆𝑏𝑎 = 𝑏𝑏𝑆𝑎) → ∀𝑐𝐴𝑑𝐴 (𝑐𝑅𝑑𝑐 = 𝑑𝑑𝑅𝑐)))
321, 31anim12d 585 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ((𝑆 Po 𝐵 ∧ ∀𝑎𝐵𝑏𝐵 (𝑎𝑆𝑏𝑎 = 𝑏𝑏𝑆𝑎)) → (𝑅 Po 𝐴 ∧ ∀𝑐𝐴𝑑𝐴 (𝑐𝑅𝑑𝑐 = 𝑑𝑑𝑅𝑐))))
33 df-so 5001 . 2 (𝑆 Or 𝐵 ↔ (𝑆 Po 𝐵 ∧ ∀𝑎𝐵𝑏𝐵 (𝑎𝑆𝑏𝑎 = 𝑏𝑏𝑆𝑎)))
34 df-so 5001 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑐𝐴𝑑𝐴 (𝑐𝑅𝑑𝑐 = 𝑑𝑑𝑅𝑐)))
3532, 33, 343imtr4g 285 1 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Or 𝐵𝑅 Or 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3o 1035   = wceq 1480  wcel 1987  wral 2907   class class class wbr 4618   Po wpo 4998   Or wor 4999  wf 5848  1-1wf1 5849  1-1-ontowf1o 5851  cfv 5852   Isom wiso 5853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-po 5000  df-so 5001  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-f1o 5859  df-fv 5860  df-isom 5861
This theorem is referenced by:  isoso  6558  isowe2  6560
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