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Theorem clsint2 35722
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 5096 . . . 4 (𝐶 ⊆ 𝒫 𝑋 𝐶𝑋)
2 elssuni 4934 . . . . . . . 8 (𝑐𝐶𝑐 𝐶)
3 sstr2 3984 . . . . . . . 8 (𝑐 𝐶 → ( 𝐶𝑋𝑐𝑋))
42, 3syl 17 . . . . . . 7 (𝑐𝐶 → ( 𝐶𝑋𝑐𝑋))
54adantl 481 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑐𝐶) → ( 𝐶𝑋𝑐𝑋))
6 intss1 4960 . . . . . . . . 9 (𝑐𝐶 𝐶𝑐)
7 clsint2.1 . . . . . . . . . 10 𝑋 = 𝐽
87clsss 22909 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑐𝑋 𝐶𝑐) → ((cls‘𝐽)‘ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
96, 8syl3an3 1162 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑐𝑋𝑐𝐶) → ((cls‘𝐽)‘ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
1093com23 1123 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑐𝐶𝑐𝑋) → ((cls‘𝐽)‘ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
11103expia 1118 . . . . . 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 451 . . . 4 ((𝐽 ∈ Top ∧ 𝐶𝑋) → (𝑐𝐶 → ((cls‘𝐽)‘ 𝐶) ⊆ ((cls‘𝐽)‘𝑐)))
141, 13sylan2b 593 . . 3 ((𝐽 ∈ Top ∧ 𝐶 ⊆ 𝒫 𝑋) → (𝑐𝐶 → ((cls‘𝐽)‘ 𝐶) ⊆ ((cls‘𝐽)‘𝑐)))
1514ralrimiv 3139 . 2 ((𝐽 ∈ Top ∧ 𝐶 ⊆ 𝒫 𝑋) → ∀𝑐𝐶 ((cls‘𝐽)‘ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
16 ssiin 5051 . 2 (((cls‘𝐽)‘ 𝐶) ⊆ 𝑐𝐶 ((cls‘𝐽)‘𝑐) ↔ ∀𝑐𝐶 ((cls‘𝐽)‘ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
1715, 16sylibr 233 1 ((𝐽 ∈ Top ∧ 𝐶 ⊆ 𝒫 𝑋) → ((cls‘𝐽)‘ 𝐶) ⊆ 𝑐𝐶 ((cls‘𝐽)‘𝑐))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  wral 3055  wss 3943  𝒫 cpw 4597   cuni 4902   cint 4943   ciin 4991  cfv 6536  Topctop 22746  clsccl 22873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-top 22747  df-cld 22874  df-cls 22876
This theorem is referenced by: (None)
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