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Theorem isosolem 6497
 Description: Lemma for isoso 6498. (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 6495 . . 3 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Po 𝐵𝑅 Po 𝐴))
2 isof1o 6473 . . . . . . . 8 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
3 f1of 6050 . . . . . . . 8 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴𝐵)
4 ffvelrn 6265 . . . . . . . . . 10 ((𝐻:𝐴𝐵𝑐𝐴) → (𝐻𝑐) ∈ 𝐵)
54ex 449 . . . . . . . . 9 (𝐻:𝐴𝐵 → (𝑐𝐴 → (𝐻𝑐) ∈ 𝐵))
6 ffvelrn 6265 . . . . . . . . . 10 ((𝐻:𝐴𝐵𝑑𝐴) → (𝐻𝑑) ∈ 𝐵)
76ex 449 . . . . . . . . 9 (𝐻:𝐴𝐵 → (𝑑𝐴 → (𝐻𝑑) ∈ 𝐵))
85, 7anim12d 584 . . . . . . . 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 444 . . . . . 6 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐𝐴𝑑𝐴)) → ((𝐻𝑐) ∈ 𝐵 ∧ (𝐻𝑑) ∈ 𝐵))
11 breq1 4586 . . . . . . . 8 (𝑎 = (𝐻𝑐) → (𝑎𝑆𝑏 ↔ (𝐻𝑐)𝑆𝑏))
12 eqeq1 2614 . . . . . . . 8 (𝑎 = (𝐻𝑐) → (𝑎 = 𝑏 ↔ (𝐻𝑐) = 𝑏))
13 breq2 4587 . . . . . . . 8 (𝑎 = (𝐻𝑐) → (𝑏𝑆𝑎𝑏𝑆(𝐻𝑐)))
1411, 12, 133orbi123d 1390 . . . . . . 7 (𝑎 = (𝐻𝑐) → ((𝑎𝑆𝑏𝑎 = 𝑏𝑏𝑆𝑎) ↔ ((𝐻𝑐)𝑆𝑏 ∨ (𝐻𝑐) = 𝑏𝑏𝑆(𝐻𝑐))))
15 breq2 4587 . . . . . . . 8 (𝑏 = (𝐻𝑑) → ((𝐻𝑐)𝑆𝑏 ↔ (𝐻𝑐)𝑆(𝐻𝑑)))
16 eqeq2 2621 . . . . . . . 8 (𝑏 = (𝐻𝑑) → ((𝐻𝑐) = 𝑏 ↔ (𝐻𝑐) = (𝐻𝑑)))
17 breq1 4586 . . . . . . . 8 (𝑏 = (𝐻𝑑) → (𝑏𝑆(𝐻𝑐) ↔ (𝐻𝑑)𝑆(𝐻𝑐)))
1815, 16, 173orbi123d 1390 . . . . . . 7 (𝑏 = (𝐻𝑑) → (((𝐻𝑐)𝑆𝑏 ∨ (𝐻𝑐) = 𝑏𝑏𝑆(𝐻𝑐)) ↔ ((𝐻𝑐)𝑆(𝐻𝑑) ∨ (𝐻𝑐) = (𝐻𝑑) ∨ (𝐻𝑑)𝑆(𝐻𝑐))))
1914, 18rspc2v 3293 . . . . . 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 6476 . . . . . 6 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐𝐴𝑑𝐴)) → (𝑐𝑅𝑑 ↔ (𝐻𝑐)𝑆(𝐻𝑑)))
22 f1of1 6049 . . . . . . . . 9 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴1-1𝐵)
232, 22syl 17 . . . . . . . 8 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1𝐵)
24 f1fveq 6420 . . . . . . . 8 ((𝐻:𝐴1-1𝐵 ∧ (𝑐𝐴𝑑𝐴)) → ((𝐻𝑐) = (𝐻𝑑) ↔ 𝑐 = 𝑑))
2523, 24sylan 487 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐𝐴𝑑𝐴)) → ((𝐻𝑐) = (𝐻𝑑) ↔ 𝑐 = 𝑑))
2625bicomd 212 . . . . . 6 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐𝐴𝑑𝐴)) → (𝑐 = 𝑑 ↔ (𝐻𝑐) = (𝐻𝑑)))
27 isorel 6476 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑐𝐴)) → (𝑑𝑅𝑐 ↔ (𝐻𝑑)𝑆(𝐻𝑐)))
2827ancom2s 840 . . . . . 6 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐𝐴𝑑𝐴)) → (𝑑𝑅𝑐 ↔ (𝐻𝑑)𝑆(𝐻𝑐)))
2921, 26, 283orbi123d 1390 . . . . 5 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐𝐴𝑑𝐴)) → ((𝑐𝑅𝑑𝑐 = 𝑑𝑑𝑅𝑐) ↔ ((𝐻𝑐)𝑆(𝐻𝑑) ∨ (𝐻𝑐) = (𝐻𝑑) ∨ (𝐻𝑑)𝑆(𝐻𝑐))))
3020, 29sylibrd 248 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐𝐴𝑑𝐴)) → (∀𝑎𝐵𝑏𝐵 (𝑎𝑆𝑏𝑎 = 𝑏𝑏𝑆𝑎) → (𝑐𝑅𝑑𝑐 = 𝑑𝑑𝑅𝑐)))
3130ralrimdvva 2957 . . 3 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑎𝐵𝑏𝐵 (𝑎𝑆𝑏𝑎 = 𝑏𝑏𝑆𝑎) → ∀𝑐𝐴𝑑𝐴 (𝑐𝑅𝑑𝑐 = 𝑑𝑑𝑅𝑐)))
321, 31anim12d 584 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ((𝑆 Po 𝐵 ∧ ∀𝑎𝐵𝑏𝐵 (𝑎𝑆𝑏𝑎 = 𝑏𝑏𝑆𝑎)) → (𝑅 Po 𝐴 ∧ ∀𝑐𝐴𝑑𝐴 (𝑐𝑅𝑑𝑐 = 𝑑𝑑𝑅𝑐))))
33 df-so 4960 . 2 (𝑆 Or 𝐵 ↔ (𝑆 Po 𝐵 ∧ ∀𝑎𝐵𝑏𝐵 (𝑎𝑆𝑏𝑎 = 𝑏𝑏𝑆𝑎)))
34 df-so 4960 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑐𝐴𝑑𝐴 (𝑐𝑅𝑑𝑐 = 𝑑𝑑𝑅𝑐)))
3532, 33, 343imtr4g 284 1 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Or 𝐵𝑅 Or 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∨ w3o 1030   = wceq 1475   ∈ wcel 1977  ∀wral 2896   class class class wbr 4583   Po wpo 4957   Or wor 4958  ⟶wf 5800  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804   Isom wiso 5805 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-f1o 5811  df-fv 5812  df-isom 5813 This theorem is referenced by:  isoso  6498  isowe2  6500
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