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Theorem clsint2 32551
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 4719 . . . 4 (𝐶 ⊆ 𝒫 𝑋 𝐶𝑋)
2 elssuni 4575 . . . . . . . 8 (𝑐𝐶𝑐 𝐶)
3 sstr2 3716 . . . . . . . 8 (𝑐 𝐶 → ( 𝐶𝑋𝑐𝑋))
42, 3syl 17 . . . . . . 7 (𝑐𝐶 → ( 𝐶𝑋𝑐𝑋))
54adantl 473 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑐𝐶) → ( 𝐶𝑋𝑐𝑋))
6 intss1 4600 . . . . . . . . 9 (𝑐𝐶 𝐶𝑐)
7 clsint2.1 . . . . . . . . . 10 𝑋 = 𝐽
87clsss 20981 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑐𝑋 𝐶𝑐) → ((cls‘𝐽)‘ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
96, 8syl3an3 1474 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑐𝑋𝑐𝐶) → ((cls‘𝐽)‘ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
1093com23 1120 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑐𝐶𝑐𝑋) → ((cls‘𝐽)‘ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
11103expia 1114 . . . . . 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 455 . . . 4 ((𝐽 ∈ Top ∧ 𝐶𝑋) → (𝑐𝐶 → ((cls‘𝐽)‘ 𝐶) ⊆ ((cls‘𝐽)‘𝑐)))
141, 13sylan2b 493 . . 3 ((𝐽 ∈ Top ∧ 𝐶 ⊆ 𝒫 𝑋) → (𝑐𝐶 → ((cls‘𝐽)‘ 𝐶) ⊆ ((cls‘𝐽)‘𝑐)))
1514ralrimiv 3067 . 2 ((𝐽 ∈ Top ∧ 𝐶 ⊆ 𝒫 𝑋) → ∀𝑐𝐶 ((cls‘𝐽)‘ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
16 ssiin 4678 . 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 383   = wceq 1596  wcel 2103  wral 3014  wss 3680  𝒫 cpw 4266   cuni 4544   cint 4583   ciin 4629  cfv 6001  Topctop 20821  clsccl 20945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-iin 4631  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-top 20822  df-cld 20946  df-cls 20948
This theorem is referenced by: (None)
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