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Theorem wemaplem1 8334
Description: Value of the lexicographic order on a sequence space. (Contributed by Stefan O'Rear, 18-Jan-2015.)
Hypothesis
Ref Expression
wemapso.t 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}
Assertion
Ref Expression
wemaplem1 ((𝑃𝑉𝑄𝑊) → (𝑃𝑇𝑄 ↔ ∃𝑎𝐴 ((𝑃𝑎)𝑆(𝑄𝑎) ∧ ∀𝑏𝐴 (𝑏𝑅𝑎 → (𝑃𝑏) = (𝑄𝑏)))))
Distinct variable groups:   𝑎,𝑏,𝑥   𝑇,𝑎,𝑏   𝑤,𝑎,𝑦,𝑧,𝑏,𝑥,𝐴   𝑃,𝑎,𝑏,𝑤,𝑥,𝑦,𝑧   𝑄,𝑎,𝑏,𝑤,𝑥,𝑦,𝑧   𝑅,𝑎,𝑏,𝑤,𝑥,𝑦,𝑧   𝑆,𝑎,𝑏,𝑤,𝑥,𝑦,𝑧
Allowed substitution hints:   𝑇(𝑥,𝑦,𝑧,𝑤)   𝑉(𝑥,𝑦,𝑧,𝑤,𝑎,𝑏)   𝑊(𝑥,𝑦,𝑧,𝑤,𝑎,𝑏)

Proof of Theorem wemaplem1
StepHypRef Expression
1 fveq1 6102 . . . . . 6 (𝑥 = 𝑃 → (𝑥𝑧) = (𝑃𝑧))
2 fveq1 6102 . . . . . 6 (𝑦 = 𝑄 → (𝑦𝑧) = (𝑄𝑧))
31, 2breqan12d 4599 . . . . 5 ((𝑥 = 𝑃𝑦 = 𝑄) → ((𝑥𝑧)𝑆(𝑦𝑧) ↔ (𝑃𝑧)𝑆(𝑄𝑧)))
4 fveq1 6102 . . . . . . . 8 (𝑥 = 𝑃 → (𝑥𝑤) = (𝑃𝑤))
5 fveq1 6102 . . . . . . . 8 (𝑦 = 𝑄 → (𝑦𝑤) = (𝑄𝑤))
64, 5eqeqan12d 2626 . . . . . . 7 ((𝑥 = 𝑃𝑦 = 𝑄) → ((𝑥𝑤) = (𝑦𝑤) ↔ (𝑃𝑤) = (𝑄𝑤)))
76imbi2d 329 . . . . . 6 ((𝑥 = 𝑃𝑦 = 𝑄) → ((𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)) ↔ (𝑤𝑅𝑧 → (𝑃𝑤) = (𝑄𝑤))))
87ralbidv 2969 . . . . 5 ((𝑥 = 𝑃𝑦 = 𝑄) → (∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)) ↔ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑃𝑤) = (𝑄𝑤))))
93, 8anbi12d 743 . . . 4 ((𝑥 = 𝑃𝑦 = 𝑄) → (((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤))) ↔ ((𝑃𝑧)𝑆(𝑄𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑃𝑤) = (𝑄𝑤)))))
109rexbidv 3034 . . 3 ((𝑥 = 𝑃𝑦 = 𝑄) → (∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤))) ↔ ∃𝑧𝐴 ((𝑃𝑧)𝑆(𝑄𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑃𝑤) = (𝑄𝑤)))))
11 fveq2 6103 . . . . . 6 (𝑧 = 𝑎 → (𝑃𝑧) = (𝑃𝑎))
12 fveq2 6103 . . . . . 6 (𝑧 = 𝑎 → (𝑄𝑧) = (𝑄𝑎))
1311, 12breq12d 4596 . . . . 5 (𝑧 = 𝑎 → ((𝑃𝑧)𝑆(𝑄𝑧) ↔ (𝑃𝑎)𝑆(𝑄𝑎)))
14 breq2 4587 . . . . . . . 8 (𝑧 = 𝑎 → (𝑤𝑅𝑧𝑤𝑅𝑎))
1514imbi1d 330 . . . . . . 7 (𝑧 = 𝑎 → ((𝑤𝑅𝑧 → (𝑃𝑤) = (𝑄𝑤)) ↔ (𝑤𝑅𝑎 → (𝑃𝑤) = (𝑄𝑤))))
1615ralbidv 2969 . . . . . 6 (𝑧 = 𝑎 → (∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑃𝑤) = (𝑄𝑤)) ↔ ∀𝑤𝐴 (𝑤𝑅𝑎 → (𝑃𝑤) = (𝑄𝑤))))
17 breq1 4586 . . . . . . . 8 (𝑤 = 𝑏 → (𝑤𝑅𝑎𝑏𝑅𝑎))
18 fveq2 6103 . . . . . . . . 9 (𝑤 = 𝑏 → (𝑃𝑤) = (𝑃𝑏))
19 fveq2 6103 . . . . . . . . 9 (𝑤 = 𝑏 → (𝑄𝑤) = (𝑄𝑏))
2018, 19eqeq12d 2625 . . . . . . . 8 (𝑤 = 𝑏 → ((𝑃𝑤) = (𝑄𝑤) ↔ (𝑃𝑏) = (𝑄𝑏)))
2117, 20imbi12d 333 . . . . . . 7 (𝑤 = 𝑏 → ((𝑤𝑅𝑎 → (𝑃𝑤) = (𝑄𝑤)) ↔ (𝑏𝑅𝑎 → (𝑃𝑏) = (𝑄𝑏))))
2221cbvralv 3147 . . . . . 6 (∀𝑤𝐴 (𝑤𝑅𝑎 → (𝑃𝑤) = (𝑄𝑤)) ↔ ∀𝑏𝐴 (𝑏𝑅𝑎 → (𝑃𝑏) = (𝑄𝑏)))
2316, 22syl6bb 275 . . . . 5 (𝑧 = 𝑎 → (∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑃𝑤) = (𝑄𝑤)) ↔ ∀𝑏𝐴 (𝑏𝑅𝑎 → (𝑃𝑏) = (𝑄𝑏))))
2413, 23anbi12d 743 . . . 4 (𝑧 = 𝑎 → (((𝑃𝑧)𝑆(𝑄𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑃𝑤) = (𝑄𝑤))) ↔ ((𝑃𝑎)𝑆(𝑄𝑎) ∧ ∀𝑏𝐴 (𝑏𝑅𝑎 → (𝑃𝑏) = (𝑄𝑏)))))
2524cbvrexv 3148 . . 3 (∃𝑧𝐴 ((𝑃𝑧)𝑆(𝑄𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑃𝑤) = (𝑄𝑤))) ↔ ∃𝑎𝐴 ((𝑃𝑎)𝑆(𝑄𝑎) ∧ ∀𝑏𝐴 (𝑏𝑅𝑎 → (𝑃𝑏) = (𝑄𝑏))))
2610, 25syl6bb 275 . 2 ((𝑥 = 𝑃𝑦 = 𝑄) → (∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤))) ↔ ∃𝑎𝐴 ((𝑃𝑎)𝑆(𝑄𝑎) ∧ ∀𝑏𝐴 (𝑏𝑅𝑎 → (𝑃𝑏) = (𝑄𝑏)))))
27 wemapso.t . 2 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}
2826, 27brabga 4914 1 ((𝑃𝑉𝑄𝑊) → (𝑃𝑇𝑄 ↔ ∃𝑎𝐴 ((𝑃𝑎)𝑆(𝑄𝑎) ∧ ∀𝑏𝐴 (𝑏𝑅𝑎 → (𝑃𝑏) = (𝑄𝑏)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wral 2896  wrex 2897   class class class wbr 4583  {copab 4642  cfv 5804
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-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-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-iota 5768  df-fv 5812
This theorem is referenced by:  wemaplem2  8335  wemaplem3  8336  wemappo  8337  wemapsolem  8338
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