Theorem neiptopuni 20934

Description: Lemma for neiptopreu 20937. (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 4168	. . . . . . . 8 ⊢ (𝑎 ∈ 𝒫 𝑋 → 𝑎 ⊆ 𝑋)
2	1	ad2antlr 763	. . . . . . 7 ⊢ (((𝑝 ∈ ∪ 𝐽 ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝 ∈ 𝑎) → 𝑎 ⊆ 𝑋)
3		simpr 477	. . . . . . 7 ⊢ (((𝑝 ∈ ∪ 𝐽 ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝 ∈ 𝑎) → 𝑝 ∈ 𝑎)
4	2, 3	sseldd 3604	. . . . . 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 4445	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)}
7	6	eleq2i 2693	. . . . . . . 8 ⊢ (𝑝 ∈ ∪ 𝐽 ↔ 𝑝 ∈ ∪ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)})
8		elunirab 4448	. . . . . . . 8 ⊢ (𝑝 ∈ ∪ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)} ↔ ∃𝑎 ∈ 𝒫 𝑋(𝑝 ∈ 𝑎 ∧ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)))
9	7, 8	bitri 264	. . . . . . 7 ⊢ (𝑝 ∈ ∪ 𝐽 ↔ ∃𝑎 ∈ 𝒫 𝑋(𝑝 ∈ 𝑎 ∧ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)))
10		simpl 473	. . . . . . . 8 ⊢ ((𝑝 ∈ 𝑎 ∧ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)) → 𝑝 ∈ 𝑎)
11	10	reximi 3011	. . . . . . 7 ⊢ (∃𝑎 ∈ 𝒫 𝑋(𝑝 ∈ 𝑎 ∧ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)) → ∃𝑎 ∈ 𝒫 𝑋𝑝 ∈ 𝑎)
12	9, 11	sylbi 207	. . . . . 6 ⊢ (𝑝 ∈ ∪ 𝐽 → ∃𝑎 ∈ 𝒫 𝑋𝑝 ∈ 𝑎)
13	4, 12	r19.29a 3078	. . . . 5 ⊢ (𝑝 ∈ ∪ 𝐽 → 𝑝 ∈ 𝑋)
14	13	a1i 11	. . . 4 ⊢ (𝜑 → (𝑝 ∈ ∪ 𝐽 → 𝑝 ∈ 𝑋))
15	14	ssrdv 3609	. . 3 ⊢ (𝜑 → ∪ 𝐽 ⊆ 𝑋)
16		ssid 3624	. . . . 5 ⊢ 𝑋 ⊆ 𝑋
17	16	a1i 11	. . . 4 ⊢ (𝜑 → 𝑋 ⊆ 𝑋)
18		neiptop.5	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑋) → 𝑋 ∈ (𝑁‘𝑝))
19	18	ralrimiva 2966	. . . 4 ⊢ (𝜑 → ∀𝑝 ∈ 𝑋 𝑋 ∈ (𝑁‘𝑝))
20	5	neipeltop 20933	. . . 4 ⊢ (𝑋 ∈ 𝐽 ↔ (𝑋 ⊆ 𝑋 ∧ ∀𝑝 ∈ 𝑋 𝑋 ∈ (𝑁‘𝑝)))
21	17, 19, 20	sylanbrc 698	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐽)
22		unissel 4468	. . 3 ⊢ ((∪ 𝐽 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → ∪ 𝐽 = 𝑋)
23	15, 21, 22	syl2anc 693	. 2 ⊢ (𝜑 → ∪ 𝐽 = 𝑋)
24	23	eqcomd 2628	1 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  {crab 2916   ⊆ wss 3574  𝒫 cpw 4158  ∪ cuni 4436  ⟶wf 5884  ‘cfv 5888  ficfi 8316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-uni 4437

This theorem is referenced by:  neiptoptop  20935  neiptopnei  20936  neiptopreu  20937