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Theorem neiptopuni 20857
 Description: Lemma for neiptopreu 20860. (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 4145 . . . . . . . 8 (𝑎 ∈ 𝒫 𝑋𝑎𝑋)
21ad2antlr 762 . . . . . . 7 (((𝑝 𝐽𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) → 𝑎𝑋)
3 simpr 477 . . . . . . 7 (((𝑝 𝐽𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) → 𝑝𝑎)
42, 3sseldd 3588 . . . . . 6 (((𝑝 𝐽𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) → 𝑝𝑋)
5 neiptop.o . . . . . . . . . 10 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)}
65unieqi 4416 . . . . . . . . 9 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)}
76eleq2i 2690 . . . . . . . 8 (𝑝 𝐽𝑝 {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)})
8 elunirab 4419 . . . . . . . 8 (𝑝 {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)} ↔ ∃𝑎 ∈ 𝒫 𝑋(𝑝𝑎 ∧ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)))
97, 8bitri 264 . . . . . . 7 (𝑝 𝐽 ↔ ∃𝑎 ∈ 𝒫 𝑋(𝑝𝑎 ∧ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)))
10 simpl 473 . . . . . . . 8 ((𝑝𝑎 ∧ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)) → 𝑝𝑎)
1110reximi 3006 . . . . . . 7 (∃𝑎 ∈ 𝒫 𝑋(𝑝𝑎 ∧ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)) → ∃𝑎 ∈ 𝒫 𝑋𝑝𝑎)
129, 11sylbi 207 . . . . . 6 (𝑝 𝐽 → ∃𝑎 ∈ 𝒫 𝑋𝑝𝑎)
134, 12r19.29a 3072 . . . . 5 (𝑝 𝐽𝑝𝑋)
1413a1i 11 . . . 4 (𝜑 → (𝑝 𝐽𝑝𝑋))
1514ssrdv 3593 . . 3 (𝜑 𝐽𝑋)
16 ssid 3608 . . . . 5 𝑋𝑋
1716a1i 11 . . . 4 (𝜑𝑋𝑋)
18 neiptop.5 . . . . 5 ((𝜑𝑝𝑋) → 𝑋 ∈ (𝑁𝑝))
1918ralrimiva 2961 . . . 4 (𝜑 → ∀𝑝𝑋 𝑋 ∈ (𝑁𝑝))
205neipeltop 20856 . . . 4 (𝑋𝐽 ↔ (𝑋𝑋 ∧ ∀𝑝𝑋 𝑋 ∈ (𝑁𝑝)))
2117, 19, 20sylanbrc 697 . . 3 (𝜑𝑋𝐽)
22 unissel 4439 . . 3 (( 𝐽𝑋𝑋𝐽) → 𝐽 = 𝑋)
2315, 21, 22syl2anc 692 . 2 (𝜑 𝐽 = 𝑋)
2423eqcomd 2627 1 (𝜑𝑋 = 𝐽)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  ∀wral 2907  ∃wrex 2908  {crab 2911   ⊆ wss 3559  𝒫 cpw 4135  ∪ cuni 4407  ⟶wf 5848  ‘cfv 5852  ficfi 8268 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4754 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-in 3566  df-ss 3573  df-nul 3897  df-pw 4137  df-uni 4408 This theorem is referenced by:  neiptoptop  20858  neiptopnei  20859  neiptopreu  20860
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