Theorem weiunlem1 36517

Description: Lemma for weiunpo 36520, weiunso 36521, weiunfr 36522, and weiunse 36523. (Contributed by Matthew House, 8-Sep-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
weiunlem1	⊢ (𝐶𝑇𝐷 ↔ ((𝐶 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝐷 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝐶)𝑅(𝐹‘𝐷) ∨ ((𝐹‘𝐶) = (𝐹‘𝐷) ∧ 𝐶⦋(𝐹‘𝐶) / 𝑥⦌𝑆𝐷))))

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

Step	Hyp	Ref	Expression
1		simpl 482	. . . . 5 ⊢ ((𝑦 = 𝐶 ∧ 𝑧 = 𝐷) → 𝑦 = 𝐶)
2	1	fveq2d 6835	. . . 4 ⊢ ((𝑦 = 𝐶 ∧ 𝑧 = 𝐷) → (𝐹‘𝑦) = (𝐹‘𝐶))
3		simpr 484	. . . . 5 ⊢ ((𝑦 = 𝐶 ∧ 𝑧 = 𝐷) → 𝑧 = 𝐷)
4	3	fveq2d 6835	. . . 4 ⊢ ((𝑦 = 𝐶 ∧ 𝑧 = 𝐷) → (𝐹‘𝑧) = (𝐹‘𝐷))
5	2, 4	breq12d 5108	. . 3 ⊢ ((𝑦 = 𝐶 ∧ 𝑧 = 𝐷) → ((𝐹‘𝑦)𝑅(𝐹‘𝑧) ↔ (𝐹‘𝐶)𝑅(𝐹‘𝐷)))
6	2, 4	eqeq12d 2749	. . . 4 ⊢ ((𝑦 = 𝐶 ∧ 𝑧 = 𝐷) → ((𝐹‘𝑦) = (𝐹‘𝑧) ↔ (𝐹‘𝐶) = (𝐹‘𝐷)))
7	2	csbeq1d 3851	. . . . 5 ⊢ ((𝑦 = 𝐶 ∧ 𝑧 = 𝐷) → ⦋(𝐹‘𝑦) / 𝑥⦌𝑆 = ⦋(𝐹‘𝐶) / 𝑥⦌𝑆)
8	1, 7, 3	breq123d 5109	. . . 4 ⊢ ((𝑦 = 𝐶 ∧ 𝑧 = 𝐷) → (𝑦⦋(𝐹‘𝑦) / 𝑥⦌𝑆𝑧 ↔ 𝐶⦋(𝐹‘𝐶) / 𝑥⦌𝑆𝐷))
9	6, 8	anbi12d 632	. . 3 ⊢ ((𝑦 = 𝐶 ∧ 𝑧 = 𝐷) → (((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦⦋(𝐹‘𝑦) / 𝑥⦌𝑆𝑧) ↔ ((𝐹‘𝐶) = (𝐹‘𝐷) ∧ 𝐶⦋(𝐹‘𝐶) / 𝑥⦌𝑆𝐷)))
10	5, 9	orbi12d 918	. 2 ⊢ ((𝑦 = 𝐶 ∧ 𝑧 = 𝐷) → (((𝐹‘𝑦)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦⦋(𝐹‘𝑦) / 𝑥⦌𝑆𝑧)) ↔ ((𝐹‘𝐶)𝑅(𝐹‘𝐷) ∨ ((𝐹‘𝐶) = (𝐹‘𝐷) ∧ 𝐶⦋(𝐹‘𝐶) / 𝑥⦌𝑆𝐷))))
11		weiun.2	. 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)))}
12	10, 11	brab2a 5714	1 ⊢ (𝐶𝑇𝐷 ↔ ((𝐶 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝐷 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝐶)𝑅(𝐹‘𝐷) ∨ ((𝐹‘𝐶) = (𝐹‘𝐷) ∧ 𝐶⦋(𝐹‘𝐶) / 𝑥⦌𝑆𝐷))))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2113  ∀wral 3049  {crab 3397  ⦋csb 3847  ∪ ciun 4943   class class class wbr 5095  {copab 5157   ↦ cmpt 5176  ‘cfv 6489  ℩crio 7311

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-xp 5627  df-iota 6445  df-fv 6497

This theorem is referenced by:  weiunpo  36520  weiunso  36521  weiunfr  36522  weiunse  36523