Theorem weiunwe 36834

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 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢))
weiun.2	⊢ 𝑇 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝑦)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦⦋(𝐹‘𝑦) / 𝑥⦌𝑆𝑧)))}

Assertion

Ref	Expression
weiunwe	⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 We 𝐵) → 𝑇 We ∪ 𝑥 ∈ 𝐴 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

Step	Hyp	Ref	Expression
1		wefr 5639	. . . 4 ⊢ (𝑆 We 𝐵 → 𝑆 Fr 𝐵)
2	1	ralimi 3101	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑆 We 𝐵 → ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵)
3		weiun.1	. . . 4 ⊢ 𝐹 = (𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢))
4		weiun.2	. . . 4 ⊢ 𝑇 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝑦)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦⦋(𝐹‘𝑦) / 𝑥⦌𝑆𝑧)))}
5	3, 4	weiunfr 36832	. . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) → 𝑇 Fr ∪ 𝑥 ∈ 𝐴 𝐵)
6	2, 5	syl3an3 1179	. 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 We 𝐵) → 𝑇 Fr ∪ 𝑥 ∈ 𝐴 𝐵)
7		weso 5640	. . . 4 ⊢ (𝑆 We 𝐵 → 𝑆 Or 𝐵)
8	7	ralimi 3101	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑆 We 𝐵 → ∀𝑥 ∈ 𝐴 𝑆 Or 𝐵)
9	3, 4	weiunso 36831	. . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Or 𝐵) → 𝑇 Or ∪ 𝑥 ∈ 𝐴 𝐵)
10	8, 9	syl3an3 1179	. 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 We 𝐵) → 𝑇 Or ∪ 𝑥 ∈ 𝐴 𝐵)
11		df-we 5604	. 2 ⊢ (𝑇 We ∪ 𝑥 ∈ 𝐴 𝐵 ↔ (𝑇 Fr ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑇 Or ∪ 𝑥 ∈ 𝐴 𝐵))
12	6, 10, 11	sylanbrc 592	1 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 We 𝐵) → 𝑇 We ∪ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 858   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  ∀wral 3078  {crab 3416  ⦋csb 3854  ∪ ciun 4951   class class class wbr 5102  {copab 5164   ↦ cmpt 5183   Or wor 5556   Fr wfr 5599   Se wse 5600   We wwe 5601  ‘cfv 6523  ℩crio 7354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-po 5557  df-so 5558  df-fr 5602  df-se 5603  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-fv 6531  df-riota 7355

This theorem is referenced by:  numiunnum  36835