Theorem neiptopuni 22855

Description: Lemma for neiptopreu 22858. (Contributed by Thierry Arnoux, 6-Jan-2018.)

Hypotheses

Ref	Expression
neiptop.o	⊢ 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)}
neiptop.0	⊢ (𝜑 → 𝑁:𝑋⟶𝒫 𝒫 𝑋)
neiptop.1	⊢ ((((𝜑 ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑏 ∈ (𝑁‘𝑝))
neiptop.2	⊢ ((𝜑 ∧ 𝑝 ∈ 𝑋) → (fi‘(𝑁‘𝑝)) ⊆ (𝑁‘𝑝))
neiptop.3	⊢ (((𝜑 ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑝 ∈ 𝑎)
neiptop.4	⊢ (((𝜑 ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → ∃𝑏 ∈ (𝑁‘𝑝)∀𝑞 ∈ 𝑏 𝑎 ∈ (𝑁‘𝑞))
neiptop.5	⊢ ((𝜑 ∧ 𝑝 ∈ 𝑋) → 𝑋 ∈ (𝑁‘𝑝))

Assertion

Ref	Expression
neiptopuni	⊢ (𝜑 → 𝑋 = ∪ 𝐽)

Distinct variable groups:   𝑝,𝑎   𝑁,𝑎   𝑋,𝑎   𝑎,𝑏,𝑝   𝐽,𝑎,𝑝   𝑋,𝑝   𝜑,𝑝

Allowed substitution hints:   𝜑(𝑞,𝑎,𝑏)   𝐽(𝑞,𝑏)   𝑁(𝑞,𝑝,𝑏)   𝑋(𝑞,𝑏)

Proof of Theorem neiptopuni

Step	Hyp	Ref	Expression
1		elpwi 4609	. . . . . . . 8 ⊢ (𝑎 ∈ 𝒫 𝑋 → 𝑎 ⊆ 𝑋)
2	1	ad2antlr 724	. . . . . . 7 ⊢ (((𝑝 ∈ ∪ 𝐽 ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝 ∈ 𝑎) → 𝑎 ⊆ 𝑋)
3		simpr 484	. . . . . . 7 ⊢ (((𝑝 ∈ ∪ 𝐽 ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝 ∈ 𝑎) → 𝑝 ∈ 𝑎)
4	2, 3	sseldd 3983	. . . . . 6 ⊢ (((𝑝 ∈ ∪ 𝐽 ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝 ∈ 𝑎) → 𝑝 ∈ 𝑋)
5		neiptop.o	. . . . . . . . . 10 ⊢ 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)}
6	5	unieqi 4921	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)}
7	6	eleq2i 2824	. . . . . . . 8 ⊢ (𝑝 ∈ ∪ 𝐽 ↔ 𝑝 ∈ ∪ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)})
8		elunirab 4924	. . . . . . . 8 ⊢ (𝑝 ∈ ∪ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)} ↔ ∃𝑎 ∈ 𝒫 𝑋(𝑝 ∈ 𝑎 ∧ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)))
9	7, 8	bitri 275	. . . . . . 7 ⊢ (𝑝 ∈ ∪ 𝐽 ↔ ∃𝑎 ∈ 𝒫 𝑋(𝑝 ∈ 𝑎 ∧ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)))
10		simpl 482	. . . . . . . 8 ⊢ ((𝑝 ∈ 𝑎 ∧ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)) → 𝑝 ∈ 𝑎)
11	10	reximi 3083	. . . . . . 7 ⊢ (∃𝑎 ∈ 𝒫 𝑋(𝑝 ∈ 𝑎 ∧ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)) → ∃𝑎 ∈ 𝒫 𝑋𝑝 ∈ 𝑎)
12	9, 11	sylbi 216	. . . . . 6 ⊢ (𝑝 ∈ ∪ 𝐽 → ∃𝑎 ∈ 𝒫 𝑋𝑝 ∈ 𝑎)
13	4, 12	r19.29a 3161	. . . . 5 ⊢ (𝑝 ∈ ∪ 𝐽 → 𝑝 ∈ 𝑋)
14	13	a1i 11	. . . 4 ⊢ (𝜑 → (𝑝 ∈ ∪ 𝐽 → 𝑝 ∈ 𝑋))
15	14	ssrdv 3988	. . 3 ⊢ (𝜑 → ∪ 𝐽 ⊆ 𝑋)
16		ssidd 4005	. . . 4 ⊢ (𝜑 → 𝑋 ⊆ 𝑋)
17		neiptop.5	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑋) → 𝑋 ∈ (𝑁‘𝑝))
18	17	ralrimiva 3145	. . . 4 ⊢ (𝜑 → ∀𝑝 ∈ 𝑋 𝑋 ∈ (𝑁‘𝑝))
19	5	neipeltop 22854	. . . 4 ⊢ (𝑋 ∈ 𝐽 ↔ (𝑋 ⊆ 𝑋 ∧ ∀𝑝 ∈ 𝑋 𝑋 ∈ (𝑁‘𝑝)))
20	16, 18, 19	sylanbrc 582	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐽)
21		unissel 4942	. . 3 ⊢ ((∪ 𝐽 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → ∪ 𝐽 = 𝑋)
22	15, 20, 21	syl2anc 583	. 2 ⊢ (𝜑 → ∪ 𝐽 = 𝑋)
23	22	eqcomd 2737	1 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  ∀wral 3060  ∃wrex 3069  {crab 3431   ⊆ wss 3948  𝒫 cpw 4602  ∪ cuni 4908  ⟶wf 6539  ‘cfv 6543  ficfi 9409

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-nul 5306

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-in 3955  df-ss 3965  df-nul 4323  df-pw 4604  df-uni 4909

This theorem is referenced by:  neiptoptop  22856  neiptopnei  22857  neiptopreu  22858