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Theorem clsint2 33138
Description: The closure of an intersection is a subset of the intersection of the closures. (Contributed by Jeff Hankins, 31-Aug-2009.)
Hypothesis
Ref Expression
clsint2.1 𝑋 = 𝐽
Assertion
Ref Expression
clsint2 ((𝐽 ∈ Top ∧ 𝐶 ⊆ 𝒫 𝑋) → ((cls‘𝐽)‘ 𝐶) ⊆ 𝑐𝐶 ((cls‘𝐽)‘𝑐))
Distinct variable groups:   𝐶,𝑐   𝐽,𝑐   𝑋,𝑐

Proof of Theorem clsint2
StepHypRef Expression
1 sspwuni 4882 . . . 4 (𝐶 ⊆ 𝒫 𝑋 𝐶𝑋)
2 elssuni 4735 . . . . . . . 8 (𝑐𝐶𝑐 𝐶)
3 sstr2 3861 . . . . . . . 8 (𝑐 𝐶 → ( 𝐶𝑋𝑐𝑋))
42, 3syl 17 . . . . . . 7 (𝑐𝐶 → ( 𝐶𝑋𝑐𝑋))
54adantl 474 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑐𝐶) → ( 𝐶𝑋𝑐𝑋))
6 intss1 4758 . . . . . . . . 9 (𝑐𝐶 𝐶𝑐)
7 clsint2.1 . . . . . . . . . 10 𝑋 = 𝐽
87clsss 21356 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑐𝑋 𝐶𝑐) → ((cls‘𝐽)‘ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
96, 8syl3an3 1145 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑐𝑋𝑐𝐶) → ((cls‘𝐽)‘ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
1093com23 1106 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑐𝐶𝑐𝑋) → ((cls‘𝐽)‘ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
11103expia 1101 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑐𝐶) → (𝑐𝑋 → ((cls‘𝐽)‘ 𝐶) ⊆ ((cls‘𝐽)‘𝑐)))
125, 11syld 47 . . . . 5 ((𝐽 ∈ Top ∧ 𝑐𝐶) → ( 𝐶𝑋 → ((cls‘𝐽)‘ 𝐶) ⊆ ((cls‘𝐽)‘𝑐)))
1312impancom 444 . . . 4 ((𝐽 ∈ Top ∧ 𝐶𝑋) → (𝑐𝐶 → ((cls‘𝐽)‘ 𝐶) ⊆ ((cls‘𝐽)‘𝑐)))
141, 13sylan2b 584 . . 3 ((𝐽 ∈ Top ∧ 𝐶 ⊆ 𝒫 𝑋) → (𝑐𝐶 → ((cls‘𝐽)‘ 𝐶) ⊆ ((cls‘𝐽)‘𝑐)))
1514ralrimiv 3125 . 2 ((𝐽 ∈ Top ∧ 𝐶 ⊆ 𝒫 𝑋) → ∀𝑐𝐶 ((cls‘𝐽)‘ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
16 ssiin 4839 . 2 (((cls‘𝐽)‘ 𝐶) ⊆ 𝑐𝐶 ((cls‘𝐽)‘𝑐) ↔ ∀𝑐𝐶 ((cls‘𝐽)‘ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
1715, 16sylibr 226 1 ((𝐽 ∈ Top ∧ 𝐶 ⊆ 𝒫 𝑋) → ((cls‘𝐽)‘ 𝐶) ⊆ 𝑐𝐶 ((cls‘𝐽)‘𝑐))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1507  wcel 2048  wral 3082  wss 3825  𝒫 cpw 4416   cuni 4706   cint 4743   ciin 4787  cfv 6182  Topctop 21195  clsccl 21320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-int 4744  df-iun 4788  df-iin 4789  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-top 21196  df-cld 21321  df-cls 21323
This theorem is referenced by: (None)
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