Theorem dfttc4lem1 36893

Description: Lemma for dfttc4 36895. (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 4167	. . . . 5 ⊢ (𝑦 = 𝐶 → (𝐴 ∩ 𝑦) = (𝐴 ∩ 𝐶))
3	2	neeq1d 3017	. . . 4 ⊢ (𝑦 = 𝐶 → ((𝐴 ∩ 𝑦) ≠ ∅ ↔ (𝐴 ∩ 𝐶) ≠ ∅))
4		ineq2 4167	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (𝑧 ∩ 𝑦) = (𝑧 ∩ 𝐶))
5	4	eqeq1d 2765	. . . . . 6 ⊢ (𝑦 = 𝐶 → ((𝑧 ∩ 𝑦) = ∅ ↔ (𝑧 ∩ 𝐶) = ∅))
6	5	imbi1d 343	. . . . 5 ⊢ (𝑦 = 𝐶 → (((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝐷) ↔ ((𝑧 ∩ 𝐶) = ∅ → 𝑧 = 𝐷)))
7	6	raleqbi1dv 3331	. . . 4 ⊢ (𝑦 = 𝐶 → (∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝐷) ↔ ∀𝑧 ∈ 𝐶 ((𝑧 ∩ 𝐶) = ∅ → 𝑧 = 𝐷)))
8	3, 7	anbi12d 641	. . 3 ⊢ (𝑦 = 𝐶 → (((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝐷)) ↔ ((𝐴 ∩ 𝐶) ≠ ∅ ∧ ∀𝑧 ∈ 𝐶 ((𝑧 ∩ 𝐶) = ∅ → 𝑧 = 𝐷))))
9	1, 8	spcev 3566	. 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 2775	. . . . . . 7 ⊢ (𝑥 = 𝐷 → (𝑧 = 𝑥 ↔ 𝑧 = 𝐷))
12	11	imbi2d 342	. . . . . 6 ⊢ (𝑥 = 𝐷 → (((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥) ↔ ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝐷)))
13	12	ralbidv 3186	. . . . 5 ⊢ (𝑥 = 𝐷 → (∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥) ↔ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝐷)))
14	13	anbi2d 639	. . . 4 ⊢ (𝑥 = 𝐷 → (((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥)) ↔ ((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝐷))))
15	14	exbidv 1942	. . 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 3642	. 2 ⊢ (𝐷 ∈ 𝐵 ↔ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝐷)))
18	9, 17	sylibr 236	1 ⊢ (((𝐴 ∩ 𝐶) ≠ ∅ ∧ ∀𝑧 ∈ 𝐶 ((𝑧 ∩ 𝐶) = ∅ → 𝑧 = 𝐷)) → 𝐷 ∈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1561  ∃wex 1800   ∈ wcel 2143  {cab 2741   ≠ wne 2958  ∀wral 3077  Vcvv 3455   ∩ cin 3904  ∅c0 4286

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1564  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-in 3912

This theorem is referenced by:  dfttc4lem2  36894