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Theorem topmeet 35297
Description: Two equivalent formulations of the meet of a collection of topologies. (Contributed by Jeff Hankins, 4-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
Assertion
Ref Expression
topmeet ((𝑋 ∈ 𝑉 ∧ 𝑆 βŠ† (TopOnβ€˜π‘‹)) β†’ (𝒫 𝑋 ∩ ∩ 𝑆) = βˆͺ {π‘˜ ∈ (TopOnβ€˜π‘‹) ∣ βˆ€π‘— ∈ 𝑆 π‘˜ βŠ† 𝑗})
Distinct variable groups:   𝑗,π‘˜,𝑆   𝑗,𝑉,π‘˜   𝑗,𝑋,π‘˜
Proof of Theorem topmeet
StepHypRef Expression
1 topmtcl 35296 . . . 4 ((𝑋 ∈ 𝑉 ∧ 𝑆 βŠ† (TopOnβ€˜π‘‹)) β†’ (𝒫 𝑋 ∩ ∩ 𝑆) ∈ (TopOnβ€˜π‘‹))
2 inss2 4230 . . . . . . 7 (𝒫 𝑋 ∩ ∩ 𝑆) βŠ† ∩ 𝑆
3 intss1 4968 . . . . . . 7 (𝑗 ∈ 𝑆 β†’ ∩ 𝑆 βŠ† 𝑗)
42, 3sstrid 3994 . . . . . 6 (𝑗 ∈ 𝑆 β†’ (𝒫 𝑋 ∩ ∩ 𝑆) βŠ† 𝑗)
54rgen 3064 . . . . 5 βˆ€π‘— ∈ 𝑆 (𝒫 𝑋 ∩ ∩ 𝑆) βŠ† 𝑗
6 sseq1 4008 . . . . . . 7 (π‘˜ = (𝒫 𝑋 ∩ ∩ 𝑆) β†’ (π‘˜ βŠ† 𝑗 ↔ (𝒫 𝑋 ∩ ∩ 𝑆) βŠ† 𝑗))
76ralbidv 3178 . . . . . 6 (π‘˜ = (𝒫 𝑋 ∩ ∩ 𝑆) β†’ (βˆ€π‘— ∈ 𝑆 π‘˜ βŠ† 𝑗 ↔ βˆ€π‘— ∈ 𝑆 (𝒫 𝑋 ∩ ∩ 𝑆) βŠ† 𝑗))
87elrab 3684 . . . . 5 ((𝒫 𝑋 ∩ ∩ 𝑆) ∈ {π‘˜ ∈ (TopOnβ€˜π‘‹) ∣ βˆ€π‘— ∈ 𝑆 π‘˜ βŠ† 𝑗} ↔ ((𝒫 𝑋 ∩ ∩ 𝑆) ∈ (TopOnβ€˜π‘‹) ∧ βˆ€π‘— ∈ 𝑆 (𝒫 𝑋 ∩ ∩ 𝑆) βŠ† 𝑗))
95, 8mpbiran2 709 . . . 4 ((𝒫 𝑋 ∩ ∩ 𝑆) ∈ {π‘˜ ∈ (TopOnβ€˜π‘‹) ∣ βˆ€π‘— ∈ 𝑆 π‘˜ βŠ† 𝑗} ↔ (𝒫 𝑋 ∩ ∩ 𝑆) ∈ (TopOnβ€˜π‘‹))
101, 9sylibr 233 . . 3 ((𝑋 ∈ 𝑉 ∧ 𝑆 βŠ† (TopOnβ€˜π‘‹)) β†’ (𝒫 𝑋 ∩ ∩ 𝑆) ∈ {π‘˜ ∈ (TopOnβ€˜π‘‹) ∣ βˆ€π‘— ∈ 𝑆 π‘˜ βŠ† 𝑗})
11 elssuni 4942 . . 3 ((𝒫 𝑋 ∩ ∩ 𝑆) ∈ {π‘˜ ∈ (TopOnβ€˜π‘‹) ∣ βˆ€π‘— ∈ 𝑆 π‘˜ βŠ† 𝑗} β†’ (𝒫 𝑋 ∩ ∩ 𝑆) βŠ† βˆͺ {π‘˜ ∈ (TopOnβ€˜π‘‹) ∣ βˆ€π‘— ∈ 𝑆 π‘˜ βŠ† 𝑗})
1210, 11syl 17 . 2 ((𝑋 ∈ 𝑉 ∧ 𝑆 βŠ† (TopOnβ€˜π‘‹)) β†’ (𝒫 𝑋 ∩ ∩ 𝑆) βŠ† βˆͺ {π‘˜ ∈ (TopOnβ€˜π‘‹) ∣ βˆ€π‘— ∈ 𝑆 π‘˜ βŠ† 𝑗})
13 toponuni 22416 . . . . . . . . 9 (π‘˜ ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ π‘˜)
14 eqimss2 4042 . . . . . . . . 9 (𝑋 = βˆͺ π‘˜ β†’ βˆͺ π‘˜ βŠ† 𝑋)
1513, 14syl 17 . . . . . . . 8 (π‘˜ ∈ (TopOnβ€˜π‘‹) β†’ βˆͺ π‘˜ βŠ† 𝑋)
16 sspwuni 5104 . . . . . . . 8 (π‘˜ βŠ† 𝒫 𝑋 ↔ βˆͺ π‘˜ βŠ† 𝑋)
1715, 16sylibr 233 . . . . . . 7 (π‘˜ ∈ (TopOnβ€˜π‘‹) β†’ π‘˜ βŠ† 𝒫 𝑋)
18173ad2ant2 1135 . . . . . 6 (((𝑋 ∈ 𝑉 ∧ 𝑆 βŠ† (TopOnβ€˜π‘‹)) ∧ π‘˜ ∈ (TopOnβ€˜π‘‹) ∧ βˆ€π‘— ∈ 𝑆 π‘˜ βŠ† 𝑗) β†’ π‘˜ βŠ† 𝒫 𝑋)
19 simp3 1139 . . . . . . 7 (((𝑋 ∈ 𝑉 ∧ 𝑆 βŠ† (TopOnβ€˜π‘‹)) ∧ π‘˜ ∈ (TopOnβ€˜π‘‹) ∧ βˆ€π‘— ∈ 𝑆 π‘˜ βŠ† 𝑗) β†’ βˆ€π‘— ∈ 𝑆 π‘˜ βŠ† 𝑗)
20 ssint 4969 . . . . . . 7 (π‘˜ βŠ† ∩ 𝑆 ↔ βˆ€π‘— ∈ 𝑆 π‘˜ βŠ† 𝑗)
2119, 20sylibr 233 . . . . . 6 (((𝑋 ∈ 𝑉 ∧ 𝑆 βŠ† (TopOnβ€˜π‘‹)) ∧ π‘˜ ∈ (TopOnβ€˜π‘‹) ∧ βˆ€π‘— ∈ 𝑆 π‘˜ βŠ† 𝑗) β†’ π‘˜ βŠ† ∩ 𝑆)
2218, 21ssind 4233 . . . . 5 (((𝑋 ∈ 𝑉 ∧ 𝑆 βŠ† (TopOnβ€˜π‘‹)) ∧ π‘˜ ∈ (TopOnβ€˜π‘‹) ∧ βˆ€π‘— ∈ 𝑆 π‘˜ βŠ† 𝑗) β†’ π‘˜ βŠ† (𝒫 𝑋 ∩ ∩ 𝑆))
23 velpw 4608 . . . . 5 (π‘˜ ∈ 𝒫 (𝒫 𝑋 ∩ ∩ 𝑆) ↔ π‘˜ βŠ† (𝒫 𝑋 ∩ ∩ 𝑆))
2422, 23sylibr 233 . . . 4 (((𝑋 ∈ 𝑉 ∧ 𝑆 βŠ† (TopOnβ€˜π‘‹)) ∧ π‘˜ ∈ (TopOnβ€˜π‘‹) ∧ βˆ€π‘— ∈ 𝑆 π‘˜ βŠ† 𝑗) β†’ π‘˜ ∈ 𝒫 (𝒫 𝑋 ∩ ∩ 𝑆))
2524rabssdv 4073 . . 3 ((𝑋 ∈ 𝑉 ∧ 𝑆 βŠ† (TopOnβ€˜π‘‹)) β†’ {π‘˜ ∈ (TopOnβ€˜π‘‹) ∣ βˆ€π‘— ∈ 𝑆 π‘˜ βŠ† 𝑗} βŠ† 𝒫 (𝒫 𝑋 ∩ ∩ 𝑆))
26 sspwuni 5104 . . 3 ({π‘˜ ∈ (TopOnβ€˜π‘‹) ∣ βˆ€π‘— ∈ 𝑆 π‘˜ βŠ† 𝑗} βŠ† 𝒫 (𝒫 𝑋 ∩ ∩ 𝑆) ↔ βˆͺ {π‘˜ ∈ (TopOnβ€˜π‘‹) ∣ βˆ€π‘— ∈ 𝑆 π‘˜ βŠ† 𝑗} βŠ† (𝒫 𝑋 ∩ ∩ 𝑆))
2725, 26sylib 217 . 2 ((𝑋 ∈ 𝑉 ∧ 𝑆 βŠ† (TopOnβ€˜π‘‹)) β†’ βˆͺ {π‘˜ ∈ (TopOnβ€˜π‘‹) ∣ βˆ€π‘— ∈ 𝑆 π‘˜ βŠ† 𝑗} βŠ† (𝒫 𝑋 ∩ ∩ 𝑆))
2812, 27eqssd 4000 1 ((𝑋 ∈ 𝑉 ∧ 𝑆 βŠ† (TopOnβ€˜π‘‹)) β†’ (𝒫 𝑋 ∩ ∩ 𝑆) = βˆͺ {π‘˜ ∈ (TopOnβ€˜π‘‹) ∣ βˆ€π‘— ∈ 𝑆 π‘˜ βŠ† 𝑗})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  {crab 3433   ∩ cin 3948   βŠ† wss 3949  π’« cpw 4603  βˆͺ cuni 4909  βˆ© cint 4951  β€˜cfv 6544  TopOnctopon 22412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-mre 17530  df-top 22396  df-topon 22413
This theorem is referenced by: (None)
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