Theorem neiptopuni 23170

Description: Lemma for neiptopreu 23173. (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 4561	. . . . . . . 8 ⊢ (𝑎 ∈ 𝒫 𝑋 → 𝑎 ⊆ 𝑋)
2	1	ad2antlr 737	. . . . . . 7 ⊢ (((𝑝 ∈ ∪ 𝐽 ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝 ∈ 𝑎) → 𝑎 ⊆ 𝑋)
3		simpr 488	. . . . . . 7 ⊢ (((𝑝 ∈ ∪ 𝐽 ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝 ∈ 𝑎) → 𝑝 ∈ 𝑎)
4	2, 3	sseldd 3937	. . . . . 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 4876	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)}
7	6	eleq2i 2853	. . . . . . . 8 ⊢ (𝑝 ∈ ∪ 𝐽 ↔ 𝑝 ∈ ∪ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)})
8		elunirab 4879	. . . . . . . 8 ⊢ (𝑝 ∈ ∪ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)} ↔ ∃𝑎 ∈ 𝒫 𝑋(𝑝 ∈ 𝑎 ∧ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)))
9	7, 8	bitri 277	. . . . . . 7 ⊢ (𝑝 ∈ ∪ 𝐽 ↔ ∃𝑎 ∈ 𝒫 𝑋(𝑝 ∈ 𝑎 ∧ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)))
10		simpl 486	. . . . . . . 8 ⊢ ((𝑝 ∈ 𝑎 ∧ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)) → 𝑝 ∈ 𝑎)
11	10	reximi 3099	. . . . . . 7 ⊢ (∃𝑎 ∈ 𝒫 𝑋(𝑝 ∈ 𝑎 ∧ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)) → ∃𝑎 ∈ 𝒫 𝑋𝑝 ∈ 𝑎)
12	9, 11	sylbi 219	. . . . . 6 ⊢ (𝑝 ∈ ∪ 𝐽 → ∃𝑎 ∈ 𝒫 𝑋𝑝 ∈ 𝑎)
13	4, 12	r19.29a 3169	. . . . 5 ⊢ (𝑝 ∈ ∪ 𝐽 → 𝑝 ∈ 𝑋)
14	13	a1i 11	. . . 4 ⊢ (𝜑 → (𝑝 ∈ ∪ 𝐽 → 𝑝 ∈ 𝑋))
15	14	ssrdv 3942	. . 3 ⊢ (𝜑 → ∪ 𝐽 ⊆ 𝑋)
16		ssidd 3959	. . . 4 ⊢ (𝜑 → 𝑋 ⊆ 𝑋)
17		neiptop.5	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑋) → 𝑋 ∈ (𝑁‘𝑝))
18	17	ralrimiva 3153	. . . 4 ⊢ (𝜑 → ∀𝑝 ∈ 𝑋 𝑋 ∈ (𝑁‘𝑝))
19	5	neipeltop 23169	. . . 4 ⊢ (𝑋 ∈ 𝐽 ↔ (𝑋 ⊆ 𝑋 ∧ ∀𝑝 ∈ 𝑋 𝑋 ∈ (𝑁‘𝑝)))
20	16, 18, 19	sylanbrc 592	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐽)
21		unissel 4897	. . 3 ⊢ ((∪ 𝐽 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → ∪ 𝐽 = 𝑋)
22	15, 20, 21	syl2anc 593	. 2 ⊢ (𝜑 → ∪ 𝐽 = 𝑋)
23	22	eqcomd 2767	1 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∀wral 3075  ∃wrex 3085  {crab 3413   ⊆ wss 3904  𝒫 cpw 4554  ∪ cuni 4864  ⟶wf 6513  ‘cfv 6517  ficfi 9353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-nul 5255

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-ss 3921  df-nul 4286  df-pw 4556  df-uni 4865

This theorem is referenced by:  neiptoptop  23171  neiptopnei  23172  neiptopreu  23173