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Theorem neiptopuni 22854
Description: Lemma for neiptopreu 22857. (Contributed by Thierry Arnoux, 6-Jan-2018.)
Hypotheses
Ref Expression
neiptop.o 𝐽 = {π‘Ž ∈ 𝒫 𝑋 ∣ βˆ€π‘ ∈ π‘Ž π‘Ž ∈ (π‘β€˜π‘)}
neiptop.0 (πœ‘ β†’ 𝑁:π‘‹βŸΆπ’« 𝒫 𝑋)
neiptop.1 ((((πœ‘ ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž βŠ† 𝑏 ∧ 𝑏 βŠ† 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) β†’ 𝑏 ∈ (π‘β€˜π‘))
neiptop.2 ((πœ‘ ∧ 𝑝 ∈ 𝑋) β†’ (fiβ€˜(π‘β€˜π‘)) βŠ† (π‘β€˜π‘))
neiptop.3 (((πœ‘ ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) β†’ 𝑝 ∈ π‘Ž)
neiptop.4 (((πœ‘ ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) β†’ βˆƒπ‘ ∈ (π‘β€˜π‘)βˆ€π‘ž ∈ 𝑏 π‘Ž ∈ (π‘β€˜π‘ž))
neiptop.5 ((πœ‘ ∧ 𝑝 ∈ 𝑋) β†’ 𝑋 ∈ (π‘β€˜π‘))
Assertion
Ref Expression
neiptopuni (πœ‘ β†’ 𝑋 = βˆͺ 𝐽)
Distinct variable groups:   𝑝,π‘Ž   𝑁,π‘Ž   𝑋,π‘Ž   π‘Ž,𝑏,𝑝   𝐽,π‘Ž,𝑝   𝑋,𝑝   πœ‘,𝑝
Allowed substitution hints:   πœ‘(π‘ž,π‘Ž,𝑏)   𝐽(π‘ž,𝑏)   𝑁(π‘ž,𝑝,𝑏)   𝑋(π‘ž,𝑏)

Proof of Theorem neiptopuni
StepHypRef Expression
1 elpwi 4608 . . . . . . . 8 (π‘Ž ∈ 𝒫 𝑋 β†’ π‘Ž βŠ† 𝑋)
21ad2antlr 723 . . . . . . 7 (((𝑝 ∈ βˆͺ 𝐽 ∧ π‘Ž ∈ 𝒫 𝑋) ∧ 𝑝 ∈ π‘Ž) β†’ π‘Ž βŠ† 𝑋)
3 simpr 483 . . . . . . 7 (((𝑝 ∈ βˆͺ 𝐽 ∧ π‘Ž ∈ 𝒫 𝑋) ∧ 𝑝 ∈ π‘Ž) β†’ 𝑝 ∈ π‘Ž)
42, 3sseldd 3982 . . . . . 6 (((𝑝 ∈ βˆͺ 𝐽 ∧ π‘Ž ∈ 𝒫 𝑋) ∧ 𝑝 ∈ π‘Ž) β†’ 𝑝 ∈ 𝑋)
5 neiptop.o . . . . . . . . . 10 𝐽 = {π‘Ž ∈ 𝒫 𝑋 ∣ βˆ€π‘ ∈ π‘Ž π‘Ž ∈ (π‘β€˜π‘)}
65unieqi 4920 . . . . . . . . 9 βˆͺ 𝐽 = βˆͺ {π‘Ž ∈ 𝒫 𝑋 ∣ βˆ€π‘ ∈ π‘Ž π‘Ž ∈ (π‘β€˜π‘)}
76eleq2i 2823 . . . . . . . 8 (𝑝 ∈ βˆͺ 𝐽 ↔ 𝑝 ∈ βˆͺ {π‘Ž ∈ 𝒫 𝑋 ∣ βˆ€π‘ ∈ π‘Ž π‘Ž ∈ (π‘β€˜π‘)})
8 elunirab 4923 . . . . . . . 8 (𝑝 ∈ βˆͺ {π‘Ž ∈ 𝒫 𝑋 ∣ βˆ€π‘ ∈ π‘Ž π‘Ž ∈ (π‘β€˜π‘)} ↔ βˆƒπ‘Ž ∈ 𝒫 𝑋(𝑝 ∈ π‘Ž ∧ βˆ€π‘ ∈ π‘Ž π‘Ž ∈ (π‘β€˜π‘)))
97, 8bitri 274 . . . . . . 7 (𝑝 ∈ βˆͺ 𝐽 ↔ βˆƒπ‘Ž ∈ 𝒫 𝑋(𝑝 ∈ π‘Ž ∧ βˆ€π‘ ∈ π‘Ž π‘Ž ∈ (π‘β€˜π‘)))
10 simpl 481 . . . . . . . 8 ((𝑝 ∈ π‘Ž ∧ βˆ€π‘ ∈ π‘Ž π‘Ž ∈ (π‘β€˜π‘)) β†’ 𝑝 ∈ π‘Ž)
1110reximi 3082 . . . . . . 7 (βˆƒπ‘Ž ∈ 𝒫 𝑋(𝑝 ∈ π‘Ž ∧ βˆ€π‘ ∈ π‘Ž π‘Ž ∈ (π‘β€˜π‘)) β†’ βˆƒπ‘Ž ∈ 𝒫 𝑋𝑝 ∈ π‘Ž)
129, 11sylbi 216 . . . . . 6 (𝑝 ∈ βˆͺ 𝐽 β†’ βˆƒπ‘Ž ∈ 𝒫 𝑋𝑝 ∈ π‘Ž)
134, 12r19.29a 3160 . . . . 5 (𝑝 ∈ βˆͺ 𝐽 β†’ 𝑝 ∈ 𝑋)
1413a1i 11 . . . 4 (πœ‘ β†’ (𝑝 ∈ βˆͺ 𝐽 β†’ 𝑝 ∈ 𝑋))
1514ssrdv 3987 . . 3 (πœ‘ β†’ βˆͺ 𝐽 βŠ† 𝑋)
16 ssidd 4004 . . . 4 (πœ‘ β†’ 𝑋 βŠ† 𝑋)
17 neiptop.5 . . . . 5 ((πœ‘ ∧ 𝑝 ∈ 𝑋) β†’ 𝑋 ∈ (π‘β€˜π‘))
1817ralrimiva 3144 . . . 4 (πœ‘ β†’ βˆ€π‘ ∈ 𝑋 𝑋 ∈ (π‘β€˜π‘))
195neipeltop 22853 . . . 4 (𝑋 ∈ 𝐽 ↔ (𝑋 βŠ† 𝑋 ∧ βˆ€π‘ ∈ 𝑋 𝑋 ∈ (π‘β€˜π‘)))
2016, 18, 19sylanbrc 581 . . 3 (πœ‘ β†’ 𝑋 ∈ 𝐽)
21 unissel 4941 . . 3 ((βˆͺ 𝐽 βŠ† 𝑋 ∧ 𝑋 ∈ 𝐽) β†’ βˆͺ 𝐽 = 𝑋)
2215, 20, 21syl2anc 582 . 2 (πœ‘ β†’ βˆͺ 𝐽 = 𝑋)
2322eqcomd 2736 1 (πœ‘ β†’ 𝑋 = βˆͺ 𝐽)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  βˆƒwrex 3068  {crab 3430   βŠ† wss 3947  π’« cpw 4601  βˆͺ cuni 4907  βŸΆwf 6538  β€˜cfv 6542  ficfi 9407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-nul 5305
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-in 3954  df-ss 3964  df-nul 4322  df-pw 4603  df-uni 4908
This theorem is referenced by:  neiptoptop  22855  neiptopnei  22856  neiptopreu  22857
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