Theorem dfttc4lem1 36716

Description: Lemma for dfttc4 36718. (Contributed by Matthew House, 6-Apr-2026.)

Hypotheses

Ref	Expression
dfttc4lem1.1	⊢ 𝐵 = {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥))}
dfttc4lem1.2	⊢ 𝐶 ∈ V
dfttc4lem1.3	⊢ 𝐷 ∈ V

Assertion

Ref	Expression
dfttc4lem1	⊢ (((𝐴 ∩ 𝐶) ≠ ∅ ∧ ∀𝑧 ∈ 𝐶 ((𝑧 ∩ 𝐶) = ∅ → 𝑧 = 𝐷)) → 𝐷 ∈ 𝐵)

Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐶,𝑧   𝑥,𝐷,𝑦,𝑧

Allowed substitution hints:   𝐴(𝑧)   𝐵(𝑥,𝑦,𝑧)   𝐶(𝑥)

Proof of Theorem dfttc4lem1

Step	Hyp	Ref	Expression
1		dfttc4lem1.2	. . 3 ⊢ 𝐶 ∈ V
2		ineq2 4155	. . . . 5 ⊢ (𝑦 = 𝐶 → (𝐴 ∩ 𝑦) = (𝐴 ∩ 𝐶))
3	2	neeq1d 2992	. . . 4 ⊢ (𝑦 = 𝐶 → ((𝐴 ∩ 𝑦) ≠ ∅ ↔ (𝐴 ∩ 𝐶) ≠ ∅))
4		ineq2 4155	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (𝑧 ∩ 𝑦) = (𝑧 ∩ 𝐶))
5	4	eqeq1d 2739	. . . . . 6 ⊢ (𝑦 = 𝐶 → ((𝑧 ∩ 𝑦) = ∅ ↔ (𝑧 ∩ 𝐶) = ∅))
6	5	imbi1d 341	. . . . 5 ⊢ (𝑦 = 𝐶 → (((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝐷) ↔ ((𝑧 ∩ 𝐶) = ∅ → 𝑧 = 𝐷)))
7	6	raleqbi1dv 3306	. . . 4 ⊢ (𝑦 = 𝐶 → (∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝐷) ↔ ∀𝑧 ∈ 𝐶 ((𝑧 ∩ 𝐶) = ∅ → 𝑧 = 𝐷)))
8	3, 7	anbi12d 633	. . 3 ⊢ (𝑦 = 𝐶 → (((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝐷)) ↔ ((𝐴 ∩ 𝐶) ≠ ∅ ∧ ∀𝑧 ∈ 𝐶 ((𝑧 ∩ 𝐶) = ∅ → 𝑧 = 𝐷))))
9	1, 8	spcev 3549	. 2 ⊢ (((𝐴 ∩ 𝐶) ≠ ∅ ∧ ∀𝑧 ∈ 𝐶 ((𝑧 ∩ 𝐶) = ∅ → 𝑧 = 𝐷)) → ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝐷)))
10		dfttc4lem1.3	. . 3 ⊢ 𝐷 ∈ V
11		eqeq2 2749	. . . . . . 7 ⊢ (𝑥 = 𝐷 → (𝑧 = 𝑥 ↔ 𝑧 = 𝐷))
12	11	imbi2d 340	. . . . . 6 ⊢ (𝑥 = 𝐷 → (((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥) ↔ ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝐷)))
13	12	ralbidv 3161	. . . . 5 ⊢ (𝑥 = 𝐷 → (∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥) ↔ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝐷)))
14	13	anbi2d 631	. . . 4 ⊢ (𝑥 = 𝐷 → (((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥)) ↔ ((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝐷))))
15	14	exbidv 1923	. . 3 ⊢ (𝑥 = 𝐷 → (∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥)) ↔ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝐷))))
16		dfttc4lem1.1	. . 3 ⊢ 𝐵 = {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥))}
17	10, 15, 16	elab2 3626	. 2 ⊢ (𝐷 ∈ 𝐵 ↔ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝐷)))
18	9, 17	sylibr 234	1 ⊢ (((𝐴 ∩ 𝐶) ≠ ∅ ∧ ∀𝑧 ∈ 𝐶 ((𝑧 ∩ 𝐶) = ∅ → 𝑧 = 𝐷)) → 𝐷 ∈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715   ≠ wne 2933  ∀wral 3052  Vcvv 3430   ∩ cin 3889  ∅c0 4274

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-in 3897

This theorem is referenced by:  dfttc4lem2  36717