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Theorem clsint2 34854
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 5064 . . . 4 (𝐶 ⊆ 𝒫 𝑋 𝐶𝑋)
2 elssuni 4902 . . . . . . . 8 (𝑐𝐶𝑐 𝐶)
3 sstr2 3955 . . . . . . . 8 (𝑐 𝐶 → ( 𝐶𝑋𝑐𝑋))
42, 3syl 17 . . . . . . 7 (𝑐𝐶 → ( 𝐶𝑋𝑐𝑋))
54adantl 483 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑐𝐶) → ( 𝐶𝑋𝑐𝑋))
6 intss1 4928 . . . . . . . . 9 (𝑐𝐶 𝐶𝑐)
7 clsint2.1 . . . . . . . . . 10 𝑋 = 𝐽
87clsss 22428 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑐𝑋 𝐶𝑐) → ((cls‘𝐽)‘ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
96, 8syl3an3 1166 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑐𝑋𝑐𝐶) → ((cls‘𝐽)‘ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
1093com23 1127 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑐𝐶𝑐𝑋) → ((cls‘𝐽)‘ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
11103expia 1122 . . . . . 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 453 . . . 4 ((𝐽 ∈ Top ∧ 𝐶𝑋) → (𝑐𝐶 → ((cls‘𝐽)‘ 𝐶) ⊆ ((cls‘𝐽)‘𝑐)))
141, 13sylan2b 595 . . 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 5019 . 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 397   = wceq 1542  wcel 2107  wral 3061  wss 3914  𝒫 cpw 4564   cuni 4869   cint 4911   ciin 4959  cfv 6500  Topctop 22265  clsccl 22392
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-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388
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 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-top 22266  df-cld 22393  df-cls 22395
This theorem is referenced by: (None)
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