Users' Mathboxes Mathbox for Matthew House < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  weiunwe Structured version   Visualization version   GIF version

Theorem weiunwe 36437
Description: A well-ordering on an indexed union can be constructed from a well-ordering on its index class and a collection of well-orderings on its members. (Contributed by Matthew House, 23-Aug-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
weiunwe ((𝑅 We 𝐴𝑅 Se 𝐴 ∧ ∀𝑥𝐴 𝑆 We 𝐵) → 𝑇 We 𝑥𝐴 𝐵)
Distinct variable groups:   𝑢,𝐴,𝑣,𝑤,𝑥   𝑦,𝐴,𝑧,𝑥   𝑢,𝐵,𝑣,𝑤   𝑦,𝐵,𝑧   𝑦,𝐹,𝑧   𝑢,𝑅,𝑣,𝑤   𝑦,𝑅,𝑧   𝑦,𝑆,𝑧
Allowed substitution hints:   𝐵(𝑥)   𝑅(𝑥)   𝑆(𝑥,𝑤,𝑣,𝑢)   𝑇(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝐹(𝑥,𝑤,𝑣,𝑢)

Proof of Theorem weiunwe
StepHypRef Expression
1 wefr 5690 . . . 4 (𝑆 We 𝐵𝑆 Fr 𝐵)
21ralimi 3089 . . 3 (∀𝑥𝐴 𝑆 We 𝐵 → ∀𝑥𝐴 𝑆 Fr 𝐵)
3 weiun.1 . . . 4 𝐹 = (𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑅𝑢))
4 weiun.2 . . . 4 𝑇 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦 𝑥𝐴 𝐵𝑧 𝑥𝐴 𝐵) ∧ ((𝐹𝑦)𝑅(𝐹𝑧) ∨ ((𝐹𝑦) = (𝐹𝑧) ∧ 𝑦(𝐹𝑦) / 𝑥𝑆𝑧)))}
53, 4weiunfr 36435 . . 3 ((𝑅 We 𝐴𝑅 Se 𝐴 ∧ ∀𝑥𝐴 𝑆 Fr 𝐵) → 𝑇 Fr 𝑥𝐴 𝐵)
62, 5syl3an3 1165 . 2 ((𝑅 We 𝐴𝑅 Se 𝐴 ∧ ∀𝑥𝐴 𝑆 We 𝐵) → 𝑇 Fr 𝑥𝐴 𝐵)
7 weso 5691 . . . 4 (𝑆 We 𝐵𝑆 Or 𝐵)
87ralimi 3089 . . 3 (∀𝑥𝐴 𝑆 We 𝐵 → ∀𝑥𝐴 𝑆 Or 𝐵)
93, 4weiunso 36434 . . 3 ((𝑅 We 𝐴𝑅 Se 𝐴 ∧ ∀𝑥𝐴 𝑆 Or 𝐵) → 𝑇 Or 𝑥𝐴 𝐵)
108, 9syl3an3 1165 . 2 ((𝑅 We 𝐴𝑅 Se 𝐴 ∧ ∀𝑥𝐴 𝑆 We 𝐵) → 𝑇 Or 𝑥𝐴 𝐵)
11 df-we 5654 . 2 (𝑇 We 𝑥𝐴 𝐵 ↔ (𝑇 Fr 𝑥𝐴 𝐵𝑇 Or 𝑥𝐴 𝐵))
126, 10, 11sylanbrc 582 1 ((𝑅 We 𝐴𝑅 Se 𝐴 ∧ ∀𝑥𝐴 𝑆 We 𝐵) → 𝑇 We 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 846  w3a 1087   = wceq 1537  wcel 2108  wral 3067  {crab 3443  csb 3921   ciun 5015   class class class wbr 5166  {copab 5228  cmpt 5249   Or wor 5606   Fr wfr 5649   Se wse 5650   We wwe 5651  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-10 2141  ax-11 2158  ax-12 2178  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-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-fv 6583  df-riota 7406
This theorem is referenced by:  numiunnum  36438
  Copyright terms: Public domain W3C validator