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Theorem weiunlem1 36430
Description: Lemma for weiunpo 36433, weiunso 36434, weiunfr 36435, and weiunse 36436. (Contributed by Matthew House, 8-Sep-2025.)
Hypotheses
Ref Expression
weiun.1 𝐹 = (𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑅𝑢))
weiun.2 𝑇 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦 𝑥𝐴 𝐵𝑧 𝑥𝐴 𝐵) ∧ ((𝐹𝑦)𝑅(𝐹𝑧) ∨ ((𝐹𝑦) = (𝐹𝑧) ∧ 𝑦(𝐹𝑦) / 𝑥𝑆𝑧)))}
Assertion
Ref Expression
weiunlem1 (𝐶𝑇𝐷 ↔ ((𝐶 𝑥𝐴 𝐵𝐷 𝑥𝐴 𝐵) ∧ ((𝐹𝐶)𝑅(𝐹𝐷) ∨ ((𝐹𝐶) = (𝐹𝐷) ∧ 𝐶(𝐹𝐶) / 𝑥𝑆𝐷))))
Distinct variable groups:   𝑢,𝐴,𝑣,𝑤,𝑥   𝑦,𝐴,𝑧,𝑥   𝑢,𝐵,𝑣,𝑤   𝑦,𝐵,𝑧   𝑦,𝐶,𝑧   𝑦,𝐷,𝑧   𝑦,𝐹,𝑧   𝑢,𝑅,𝑣,𝑤   𝑦,𝑅,𝑧   𝑦,𝑆,𝑧
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥,𝑤,𝑣,𝑢)   𝐷(𝑥,𝑤,𝑣,𝑢)   𝑅(𝑥)   𝑆(𝑥,𝑤,𝑣,𝑢)   𝑇(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝐹(𝑥,𝑤,𝑣,𝑢)

Proof of Theorem weiunlem1
StepHypRef Expression
1 simpl 482 . . . . 5 ((𝑦 = 𝐶𝑧 = 𝐷) → 𝑦 = 𝐶)
21fveq2d 6926 . . . 4 ((𝑦 = 𝐶𝑧 = 𝐷) → (𝐹𝑦) = (𝐹𝐶))
3 simpr 484 . . . . 5 ((𝑦 = 𝐶𝑧 = 𝐷) → 𝑧 = 𝐷)
43fveq2d 6926 . . . 4 ((𝑦 = 𝐶𝑧 = 𝐷) → (𝐹𝑧) = (𝐹𝐷))
52, 4breq12d 5179 . . 3 ((𝑦 = 𝐶𝑧 = 𝐷) → ((𝐹𝑦)𝑅(𝐹𝑧) ↔ (𝐹𝐶)𝑅(𝐹𝐷)))
62, 4eqeq12d 2756 . . . 4 ((𝑦 = 𝐶𝑧 = 𝐷) → ((𝐹𝑦) = (𝐹𝑧) ↔ (𝐹𝐶) = (𝐹𝐷)))
72csbeq1d 3925 . . . . 5 ((𝑦 = 𝐶𝑧 = 𝐷) → (𝐹𝑦) / 𝑥𝑆 = (𝐹𝐶) / 𝑥𝑆)
81, 7, 3breq123d 5180 . . . 4 ((𝑦 = 𝐶𝑧 = 𝐷) → (𝑦(𝐹𝑦) / 𝑥𝑆𝑧𝐶(𝐹𝐶) / 𝑥𝑆𝐷))
96, 8anbi12d 631 . . 3 ((𝑦 = 𝐶𝑧 = 𝐷) → (((𝐹𝑦) = (𝐹𝑧) ∧ 𝑦(𝐹𝑦) / 𝑥𝑆𝑧) ↔ ((𝐹𝐶) = (𝐹𝐷) ∧ 𝐶(𝐹𝐶) / 𝑥𝑆𝐷)))
105, 9orbi12d 917 . 2 ((𝑦 = 𝐶𝑧 = 𝐷) → (((𝐹𝑦)𝑅(𝐹𝑧) ∨ ((𝐹𝑦) = (𝐹𝑧) ∧ 𝑦(𝐹𝑦) / 𝑥𝑆𝑧)) ↔ ((𝐹𝐶)𝑅(𝐹𝐷) ∨ ((𝐹𝐶) = (𝐹𝐷) ∧ 𝐶(𝐹𝐶) / 𝑥𝑆𝐷))))
11 weiun.2 . 2 𝑇 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦 𝑥𝐴 𝐵𝑧 𝑥𝐴 𝐵) ∧ ((𝐹𝑦)𝑅(𝐹𝑧) ∨ ((𝐹𝑦) = (𝐹𝑧) ∧ 𝑦(𝐹𝑦) / 𝑥𝑆𝑧)))}
1210, 11brab2a 5793 1 (𝐶𝑇𝐷 ↔ ((𝐶 𝑥𝐴 𝐵𝐷 𝑥𝐴 𝐵) ∧ ((𝐹𝐶)𝑅(𝐹𝐷) ∨ ((𝐹𝐶) = (𝐹𝐷) ∧ 𝐶(𝐹𝐶) / 𝑥𝑆𝐷))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wa 395  wo 846   = wceq 1537  wcel 2108  wral 3067  {crab 3443  csb 3921   ciun 5015   class class class wbr 5166  {copab 5228  cmpt 5249  cfv 6575  crio 7405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-xp 5706  df-iota 6527  df-fv 6583
This theorem is referenced by:  weiunpo  36433  weiunso  36434  weiunfr  36435  weiunse  36436
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