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Theorem riincld 22992
Description: An indexed relative intersection of closed sets is closed. (Contributed by Stefan O'Rear, 22-Feb-2015.)
Hypothesis
Ref Expression
clscld.1 𝑋 = 𝐽
Assertion
Ref Expression
riincld ((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → (𝑋 𝑥𝐴 𝐵) ∈ (Clsd‘𝐽))
Distinct variable groups:   𝑥,𝐽   𝑥,𝑋   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem riincld
StepHypRef Expression
1 riin0 5086 . . . 4 (𝐴 = ∅ → (𝑋 𝑥𝐴 𝐵) = 𝑋)
21adantl 480 . . 3 (((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 = ∅) → (𝑋 𝑥𝐴 𝐵) = 𝑋)
3 clscld.1 . . . . 5 𝑋 = 𝐽
43topcld 22983 . . . 4 (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
54ad2antrr 724 . . 3 (((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 = ∅) → 𝑋 ∈ (Clsd‘𝐽))
62, 5eqeltrd 2825 . 2 (((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 = ∅) → (𝑋 𝑥𝐴 𝐵) ∈ (Clsd‘𝐽))
74ad2antrr 724 . . 3 (((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 ≠ ∅) → 𝑋 ∈ (Clsd‘𝐽))
8 simpr 483 . . . 4 (((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 ≠ ∅) → 𝐴 ≠ ∅)
9 simplr 767 . . . 4 (((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 ≠ ∅) → ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽))
10 iincld 22987 . . . 4 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 𝐵 ∈ (Clsd‘𝐽))
118, 9, 10syl2anc 582 . . 3 (((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 ≠ ∅) → 𝑥𝐴 𝐵 ∈ (Clsd‘𝐽))
12 incld 22991 . . 3 ((𝑋 ∈ (Clsd‘𝐽) ∧ 𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → (𝑋 𝑥𝐴 𝐵) ∈ (Clsd‘𝐽))
137, 11, 12syl2anc 582 . 2 (((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 ≠ ∅) → (𝑋 𝑥𝐴 𝐵) ∈ (Clsd‘𝐽))
146, 13pm2.61dane 3018 1 ((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → (𝑋 𝑥𝐴 𝐵) ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wne 2929  wral 3050  cin 3943  c0 4322   cuni 4909   ciin 4998  cfv 6549  Topctop 22839  Clsdccld 22964
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 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-iin 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6501  df-fun 6551  df-fn 6552  df-fv 6557  df-top 22840  df-cld 22967
This theorem is referenced by:  ptcld  23561  csscld  25221
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