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Theorem weiunwe 36384
Description: A well-ordering on an indexed union can be constructed from a well-ordering on its index set and a collection of well-orderings on its members. (Contributed by Matthew House, 23-Aug-2025.)
Hypotheses
Ref Expression
weiunwe.1 𝐹 = (𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑅𝑢))
weiunwe.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:   𝑥,𝐴,𝑦,𝑧   𝑢,𝐴,𝑣,𝑤,𝑥   𝑦,𝐵,𝑧   𝑢,𝐵,𝑣,𝑤   𝑦,𝐹,𝑧   𝑦,𝑅,𝑧   𝑢,𝑅,𝑣,𝑤   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 5689 . . . 4 (𝑆 We 𝐵𝑆 Fr 𝐵)
21ralimi 3085 . . 3 (∀𝑥𝐴 𝑆 We 𝐵 → ∀𝑥𝐴 𝑆 Fr 𝐵)
3 weiunwe.1 . . . 4 𝐹 = (𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑅𝑢))
4 weiunwe.2 . . . 4 𝑇 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦 𝑥𝐴 𝐵𝑧 𝑥𝐴 𝐵) ∧ ((𝐹𝑦)𝑅(𝐹𝑧) ∨ ((𝐹𝑦) = (𝐹𝑧) ∧ 𝑦(𝐹𝑦) / 𝑥𝑆𝑧)))}
53, 4weiunfr 36382 . . 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 5690 . . . 4 (𝑆 We 𝐵𝑆 Or 𝐵)
87ralimi 3085 . . 3 (∀𝑥𝐴 𝑆 We 𝐵 → ∀𝑥𝐴 𝑆 Or 𝐵)
93, 4weiunso 36379 . . 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 5656 . 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 2103  wral 3063  {crab 3438  csb 3915   ciun 5019   class class class wbr 5169  {copab 5231  cmpt 5252   Or wor 5610   Fr wfr 5651   Se wse 5652   We wwe 5653  cfv 6572  crio 7400
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 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450
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 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-rmo 3383  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5021  df-br 5170  df-opab 5232  df-mpt 5253  df-id 5597  df-po 5611  df-so 5612  df-fr 5654  df-se 5655  df-we 5656  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-fv 6580  df-riota 7401
This theorem is referenced by:  numiunnum  36385
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