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Theorem clsint2 31958
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 4582 . . . 4 (𝐶 ⊆ 𝒫 𝑋 𝐶𝑋)
2 elssuni 4438 . . . . . . . 8 (𝑐𝐶𝑐 𝐶)
3 sstr2 3595 . . . . . . . 8 (𝑐 𝐶 → ( 𝐶𝑋𝑐𝑋))
42, 3syl 17 . . . . . . 7 (𝑐𝐶 → ( 𝐶𝑋𝑐𝑋))
54adantl 482 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑐𝐶) → ( 𝐶𝑋𝑐𝑋))
6 intss1 4462 . . . . . . . . 9 (𝑐𝐶 𝐶𝑐)
7 clsint2.1 . . . . . . . . . 10 𝑋 = 𝐽
87clsss 20763 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑐𝑋 𝐶𝑐) → ((cls‘𝐽)‘ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
96, 8syl3an3 1358 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑐𝑋𝑐𝐶) → ((cls‘𝐽)‘ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
1093com23 1268 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑐𝐶𝑐𝑋) → ((cls‘𝐽)‘ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
11103expia 1264 . . . . . 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 456 . . . 4 ((𝐽 ∈ Top ∧ 𝐶𝑋) → (𝑐𝐶 → ((cls‘𝐽)‘ 𝐶) ⊆ ((cls‘𝐽)‘𝑐)))
141, 13sylan2b 492 . . 3 ((𝐽 ∈ Top ∧ 𝐶 ⊆ 𝒫 𝑋) → (𝑐𝐶 → ((cls‘𝐽)‘ 𝐶) ⊆ ((cls‘𝐽)‘𝑐)))
1514ralrimiv 2964 . 2 ((𝐽 ∈ Top ∧ 𝐶 ⊆ 𝒫 𝑋) → ∀𝑐𝐶 ((cls‘𝐽)‘ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
16 ssiin 4541 . 2 (((cls‘𝐽)‘ 𝐶) ⊆ 𝑐𝐶 ((cls‘𝐽)‘𝑐) ↔ ∀𝑐𝐶 ((cls‘𝐽)‘ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
1715, 16sylibr 224 1 ((𝐽 ∈ Top ∧ 𝐶 ⊆ 𝒫 𝑋) → ((cls‘𝐽)‘ 𝐶) ⊆ 𝑐𝐶 ((cls‘𝐽)‘𝑐))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  wral 2912  wss 3560  𝒫 cpw 4135   cuni 4407   cint 4445   ciin 4491  cfv 5850  Topctop 20612  clsccl 20727
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-top 20616  df-cld 20728  df-cls 20730
This theorem is referenced by: (None)
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