Theorem neiptopuni 20744

Description: Lemma for neiptopreu 20747. (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 4117	. . . . . . . 8 ⊢ (𝑎 ∈ 𝒫 𝑋 → 𝑎 ⊆ 𝑋)
2	1	ad2antlr 759	. . . . . . 7 ⊢ (((𝑝 ∈ ∪ 𝐽 ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝 ∈ 𝑎) → 𝑎 ⊆ 𝑋)
3		simpr 476	. . . . . . 7 ⊢ (((𝑝 ∈ ∪ 𝐽 ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝 ∈ 𝑎) → 𝑝 ∈ 𝑎)
4	2, 3	sseldd 3569	. . . . . 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 4381	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)}
7	6	eleq2i 2680	. . . . . . . 8 ⊢ (𝑝 ∈ ∪ 𝐽 ↔ 𝑝 ∈ ∪ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)})
8		elunirab 4384	. . . . . . . 8 ⊢ (𝑝 ∈ ∪ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)} ↔ ∃𝑎 ∈ 𝒫 𝑋(𝑝 ∈ 𝑎 ∧ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)))
9	7, 8	bitri 263	. . . . . . 7 ⊢ (𝑝 ∈ ∪ 𝐽 ↔ ∃𝑎 ∈ 𝒫 𝑋(𝑝 ∈ 𝑎 ∧ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)))
10		simpl 472	. . . . . . . 8 ⊢ ((𝑝 ∈ 𝑎 ∧ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)) → 𝑝 ∈ 𝑎)
11	10	reximi 2994	. . . . . . 7 ⊢ (∃𝑎 ∈ 𝒫 𝑋(𝑝 ∈ 𝑎 ∧ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)) → ∃𝑎 ∈ 𝒫 𝑋𝑝 ∈ 𝑎)
12	9, 11	sylbi 206	. . . . . 6 ⊢ (𝑝 ∈ ∪ 𝐽 → ∃𝑎 ∈ 𝒫 𝑋𝑝 ∈ 𝑎)
13	4, 12	r19.29a 3060	. . . . 5 ⊢ (𝑝 ∈ ∪ 𝐽 → 𝑝 ∈ 𝑋)
14	13	a1i 11	. . . 4 ⊢ (𝜑 → (𝑝 ∈ ∪ 𝐽 → 𝑝 ∈ 𝑋))
15	14	ssrdv 3574	. . 3 ⊢ (𝜑 → ∪ 𝐽 ⊆ 𝑋)
16		ssid 3587	. . . . 5 ⊢ 𝑋 ⊆ 𝑋
17	16	a1i 11	. . . 4 ⊢ (𝜑 → 𝑋 ⊆ 𝑋)
18		neiptop.5	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑋) → 𝑋 ∈ (𝑁‘𝑝))
19	18	ralrimiva 2949	. . . 4 ⊢ (𝜑 → ∀𝑝 ∈ 𝑋 𝑋 ∈ (𝑁‘𝑝))
20	5	neipeltop 20743	. . . 4 ⊢ (𝑋 ∈ 𝐽 ↔ (𝑋 ⊆ 𝑋 ∧ ∀𝑝 ∈ 𝑋 𝑋 ∈ (𝑁‘𝑝)))
21	17, 19, 20	sylanbrc 695	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐽)
22		unissel 4404	. . 3 ⊢ ((∪ 𝐽 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → ∪ 𝐽 = 𝑋)
23	15, 21, 22	syl2anc 691	. 2 ⊢ (𝜑 → ∪ 𝐽 = 𝑋)
24	23	eqcomd 2616	1 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372  ⟶wf 5800  ‘cfv 5804  ficfi 8199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875  df-pw 4110  df-uni 4373

This theorem is referenced by:  neiptoptop  20745  neiptopnei  20746  neiptopreu  20747