Theorem neisspw 22610

Description: The neighborhoods of any set are subsets of the base set. (Contributed by Stefan O'Rear, 6-Aug-2015.)

Hypothesis

Ref	Expression
neifval.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
neisspw	⊢ (𝐽 ∈ Top → ((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋)

Proof of Theorem neisspw

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		neifval.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
2	1	neii1 22609	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑣 ∈ ((nei‘𝐽)‘𝑆)) → 𝑣 ⊆ 𝑋)
3		velpw 4607	. . . 4 ⊢ (𝑣 ∈ 𝒫 𝑋 ↔ 𝑣 ⊆ 𝑋)
4	2, 3	sylibr 233	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑣 ∈ ((nei‘𝐽)‘𝑆)) → 𝑣 ∈ 𝒫 𝑋)
5	4	ex 413	. 2 ⊢ (𝐽 ∈ Top → (𝑣 ∈ ((nei‘𝐽)‘𝑆) → 𝑣 ∈ 𝒫 𝑋))
6	5	ssrdv 3988	1 ⊢ (𝐽 ∈ Top → ((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ⊆ wss 3948  𝒫 cpw 4602  ∪ cuni 4908  ‘cfv 6543  Topctop 22394  neicnei 22600

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-top 22395  df-nei 22601

This theorem is referenced by:  hausflim  23484  flimclslem  23487  fclsfnflim  23530