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Theorem weiunwe 36452
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 5630 . . . 4 (𝑆 We 𝐵𝑆 Fr 𝐵)
21ralimi 3067 . . 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 36450 . . 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 5631 . . . 4 (𝑆 We 𝐵𝑆 Or 𝐵)
87ralimi 3067 . . 3 (∀𝑥𝐴 𝑆 We 𝐵 → ∀𝑥𝐴 𝑆 Or 𝐵)
93, 4weiunso 36449 . . 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 5595 . 2 (𝑇 We 𝑥𝐴 𝐵 ↔ (𝑇 Fr 𝑥𝐴 𝐵𝑇 Or 𝑥𝐴 𝐵))
126, 10, 11sylanbrc 583 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 847  w3a 1086   = wceq 1540  wcel 2109  wral 3045  {crab 3408  csb 3864   ciun 4957   class class class wbr 5109  {copab 5171  cmpt 5190   Or wor 5547   Fr wfr 5590   Se wse 5591   We wwe 5592  cfv 6513  crio 7345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-po 5548  df-so 5549  df-fr 5593  df-se 5594  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-fv 6521  df-riota 7346
This theorem is referenced by:  numiunnum  36453
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