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Theorem riincld 21653
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 4970 . . . 4 (𝐴 = ∅ → (𝑋 𝑥𝐴 𝐵) = 𝑋)
21adantl 485 . . 3 (((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 = ∅) → (𝑋 𝑥𝐴 𝐵) = 𝑋)
3 clscld.1 . . . . 5 𝑋 = 𝐽
43topcld 21644 . . . 4 (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
54ad2antrr 725 . . 3 (((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 = ∅) → 𝑋 ∈ (Clsd‘𝐽))
62, 5eqeltrd 2893 . 2 (((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 = ∅) → (𝑋 𝑥𝐴 𝐵) ∈ (Clsd‘𝐽))
74ad2antrr 725 . . 3 (((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 ≠ ∅) → 𝑋 ∈ (Clsd‘𝐽))
8 simpr 488 . . . 4 (((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 ≠ ∅) → 𝐴 ≠ ∅)
9 simplr 768 . . . 4 (((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 ≠ ∅) → ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽))
10 iincld 21648 . . . 4 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 𝐵 ∈ (Clsd‘𝐽))
118, 9, 10syl2anc 587 . . 3 (((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 ≠ ∅) → 𝑥𝐴 𝐵 ∈ (Clsd‘𝐽))
12 incld 21652 . . 3 ((𝑋 ∈ (Clsd‘𝐽) ∧ 𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → (𝑋 𝑥𝐴 𝐵) ∈ (Clsd‘𝐽))
137, 11, 12syl2anc 587 . 2 (((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 ≠ ∅) → (𝑋 𝑥𝐴 𝐵) ∈ (Clsd‘𝐽))
146, 13pm2.61dane 3077 1 ((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → (𝑋 𝑥𝐴 𝐵) ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  wne 2990  wral 3109  cin 3883  c0 4246   cuni 4803   ciin 4885  cfv 6328  Topctop 21502  Clsdccld 21625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fn 6331  df-fv 6336  df-top 21503  df-cld 21628
This theorem is referenced by:  ptcld  22222  csscld  23857
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