Theorem riincld 23023

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 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → (𝑋 ∩ ∩ 𝑥 ∈ 𝐴 𝐵) ∈ (Clsd‘𝐽))

Distinct variable groups:   𝑥,𝐽   𝑥,𝑋   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem riincld

Step	Hyp	Ref	Expression
1		riin0 5025	. . . 4 ⊢ (𝐴 = ∅ → (𝑋 ∩ ∩ 𝑥 ∈ 𝐴 𝐵) = 𝑋)
2	1	adantl 481	. . 3 ⊢ (((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 = ∅) → (𝑋 ∩ ∩ 𝑥 ∈ 𝐴 𝐵) = 𝑋)
3		clscld.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
4	3	topcld 23014	. . . 4 ⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
5	4	ad2antrr 727	. . 3 ⊢ (((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 = ∅) → 𝑋 ∈ (Clsd‘𝐽))
6	2, 5	eqeltrd 2837	. 2 ⊢ (((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 = ∅) → (𝑋 ∩ ∩ 𝑥 ∈ 𝐴 𝐵) ∈ (Clsd‘𝐽))
7	4	ad2antrr 727	. . 3 ⊢ (((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 ≠ ∅) → 𝑋 ∈ (Clsd‘𝐽))
8		simpr 484	. . . 4 ⊢ (((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 ≠ ∅) → 𝐴 ≠ ∅)
9		simplr 769	. . . 4 ⊢ (((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 ≠ ∅) → ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽))
10		iincld 23018	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽))
11	8, 9, 10	syl2anc 585	. . 3 ⊢ (((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 ≠ ∅) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽))
12		incld 23022	. . 3 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ ∩ 𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → (𝑋 ∩ ∩ 𝑥 ∈ 𝐴 𝐵) ∈ (Clsd‘𝐽))
13	7, 11, 12	syl2anc 585	. 2 ⊢ (((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 ≠ ∅) → (𝑋 ∩ ∩ 𝑥 ∈ 𝐴 𝐵) ∈ (Clsd‘𝐽))
14	6, 13	pm2.61dane 3020	1 ⊢ ((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → (𝑋 ∩ ∩ 𝑥 ∈ 𝐴 𝐵) ∈ (Clsd‘𝐽))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052   ∩ cin 3889  ∅c0 4274  ∪ cuni 4851  ∩ ciin 4935  ‘cfv 6494  Topctop 22872  Clsdccld 22995

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-iota 6450  df-fun 6496  df-fn 6497  df-fv 6502  df-top 22873  df-cld 22998

This theorem is referenced by:  ptcld  23592  csscld  25230