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Theorem neiptopuni 22481
Description: Lemma for neiptopreu 22484. (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 4567 . . . . . . . 8 (𝑎 ∈ 𝒫 𝑋𝑎𝑋)
21ad2antlr 725 . . . . . . 7 (((𝑝 𝐽𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) → 𝑎𝑋)
3 simpr 485 . . . . . . 7 (((𝑝 𝐽𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) → 𝑝𝑎)
42, 3sseldd 3945 . . . . . 6 (((𝑝 𝐽𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) → 𝑝𝑋)
5 neiptop.o . . . . . . . . . 10 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)}
65unieqi 4878 . . . . . . . . 9 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)}
76eleq2i 2829 . . . . . . . 8 (𝑝 𝐽𝑝 {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)})
8 elunirab 4881 . . . . . . . 8 (𝑝 {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)} ↔ ∃𝑎 ∈ 𝒫 𝑋(𝑝𝑎 ∧ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)))
97, 8bitri 274 . . . . . . 7 (𝑝 𝐽 ↔ ∃𝑎 ∈ 𝒫 𝑋(𝑝𝑎 ∧ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)))
10 simpl 483 . . . . . . . 8 ((𝑝𝑎 ∧ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)) → 𝑝𝑎)
1110reximi 3087 . . . . . . 7 (∃𝑎 ∈ 𝒫 𝑋(𝑝𝑎 ∧ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)) → ∃𝑎 ∈ 𝒫 𝑋𝑝𝑎)
129, 11sylbi 216 . . . . . 6 (𝑝 𝐽 → ∃𝑎 ∈ 𝒫 𝑋𝑝𝑎)
134, 12r19.29a 3159 . . . . 5 (𝑝 𝐽𝑝𝑋)
1413a1i 11 . . . 4 (𝜑 → (𝑝 𝐽𝑝𝑋))
1514ssrdv 3950 . . 3 (𝜑 𝐽𝑋)
16 ssidd 3967 . . . 4 (𝜑𝑋𝑋)
17 neiptop.5 . . . . 5 ((𝜑𝑝𝑋) → 𝑋 ∈ (𝑁𝑝))
1817ralrimiva 3143 . . . 4 (𝜑 → ∀𝑝𝑋 𝑋 ∈ (𝑁𝑝))
195neipeltop 22480 . . . 4 (𝑋𝐽 ↔ (𝑋𝑋 ∧ ∀𝑝𝑋 𝑋 ∈ (𝑁𝑝)))
2016, 18, 19sylanbrc 583 . . 3 (𝜑𝑋𝐽)
21 unissel 4899 . . 3 (( 𝐽𝑋𝑋𝐽) → 𝐽 = 𝑋)
2215, 20, 21syl2anc 584 . 2 (𝜑 𝐽 = 𝑋)
2322eqcomd 2742 1 (𝜑𝑋 = 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  wrex 3073  {crab 3407  wss 3910  𝒫 cpw 4560   cuni 4865  wf 6492  cfv 6496  ficfi 9346
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-nul 5263
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-in 3917  df-ss 3927  df-nul 4283  df-pw 4562  df-uni 4866
This theorem is referenced by:  neiptoptop  22482  neiptopnei  22483  neiptopreu  22484
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