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Theorem neiptopuni 21773
 Description: Lemma for neiptopreu 21776. (Contributed by Thierry Arnoux, 6-Jan-2018.)
Hypotheses
Ref Expression
neiptop.o 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)}
neiptop.0 (𝜑𝑁:𝑋⟶𝒫 𝒫 𝑋)
neiptop.1 ((((𝜑𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑏 ∈ (𝑁𝑝))
neiptop.2 ((𝜑𝑝𝑋) → (fi‘(𝑁𝑝)) ⊆ (𝑁𝑝))
neiptop.3 (((𝜑𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑝𝑎)
neiptop.4 (((𝜑𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → ∃𝑏 ∈ (𝑁𝑝)∀𝑞𝑏 𝑎 ∈ (𝑁𝑞))
neiptop.5 ((𝜑𝑝𝑋) → 𝑋 ∈ (𝑁𝑝))
Assertion
Ref Expression
neiptopuni (𝜑𝑋 = 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
StepHypRef Expression
1 elpwi 4508 . . . . . . . 8 (𝑎 ∈ 𝒫 𝑋𝑎𝑋)
21ad2antlr 726 . . . . . . 7 (((𝑝 𝐽𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) → 𝑎𝑋)
3 simpr 488 . . . . . . 7 (((𝑝 𝐽𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) → 𝑝𝑎)
42, 3sseldd 3917 . . . . . 6 (((𝑝 𝐽𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) → 𝑝𝑋)
5 neiptop.o . . . . . . . . . 10 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)}
65unieqi 4816 . . . . . . . . 9 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)}
76eleq2i 2881 . . . . . . . 8 (𝑝 𝐽𝑝 {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)})
8 elunirab 4819 . . . . . . . 8 (𝑝 {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)} ↔ ∃𝑎 ∈ 𝒫 𝑋(𝑝𝑎 ∧ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)))
97, 8bitri 278 . . . . . . 7 (𝑝 𝐽 ↔ ∃𝑎 ∈ 𝒫 𝑋(𝑝𝑎 ∧ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)))
10 simpl 486 . . . . . . . 8 ((𝑝𝑎 ∧ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)) → 𝑝𝑎)
1110reximi 3206 . . . . . . 7 (∃𝑎 ∈ 𝒫 𝑋(𝑝𝑎 ∧ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)) → ∃𝑎 ∈ 𝒫 𝑋𝑝𝑎)
129, 11sylbi 220 . . . . . 6 (𝑝 𝐽 → ∃𝑎 ∈ 𝒫 𝑋𝑝𝑎)
134, 12r19.29a 3248 . . . . 5 (𝑝 𝐽𝑝𝑋)
1413a1i 11 . . . 4 (𝜑 → (𝑝 𝐽𝑝𝑋))
1514ssrdv 3922 . . 3 (𝜑 𝐽𝑋)
16 ssidd 3939 . . . 4 (𝜑𝑋𝑋)
17 neiptop.5 . . . . 5 ((𝜑𝑝𝑋) → 𝑋 ∈ (𝑁𝑝))
1817ralrimiva 3149 . . . 4 (𝜑 → ∀𝑝𝑋 𝑋 ∈ (𝑁𝑝))
195neipeltop 21772 . . . 4 (𝑋𝐽 ↔ (𝑋𝑋 ∧ ∀𝑝𝑋 𝑋 ∈ (𝑁𝑝)))
2016, 18, 19sylanbrc 586 . . 3 (𝜑𝑋𝐽)
21 unissel 4834 . . 3 (( 𝐽𝑋𝑋𝐽) → 𝐽 = 𝑋)
2215, 20, 21syl2anc 587 . 2 (𝜑 𝐽 = 𝑋)
2322eqcomd 2804 1 (𝜑𝑋 = 𝐽)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  {crab 3110   ⊆ wss 3882  𝒫 cpw 4499  ∪ cuni 4803  ⟶wf 6325  ‘cfv 6329  ficfi 8873 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-nul 5177 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3885  df-in 3889  df-ss 3899  df-nul 4246  df-pw 4501  df-uni 4804 This theorem is referenced by:  neiptoptop  21774  neiptopnei  21775  neiptopreu  21776
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