Theorem neisspw 23043

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 23042	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑣 ∈ ((nei‘𝐽)‘𝑆)) → 𝑣 ⊆ 𝑋)
3		velpw 4580	. . . 4 ⊢ (𝑣 ∈ 𝒫 𝑋 ↔ 𝑣 ⊆ 𝑋)
4	2, 3	sylibr 234	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑣 ∈ ((nei‘𝐽)‘𝑆)) → 𝑣 ∈ 𝒫 𝑋)
5	4	ex 412	. 2 ⊢ (𝐽 ∈ Top → (𝑣 ∈ ((nei‘𝐽)‘𝑆) → 𝑣 ∈ 𝒫 𝑋))
6	5	ssrdv 3964	1 ⊢ (𝐽 ∈ Top → ((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ⊆ wss 3926  𝒫 cpw 4575  ∪ cuni 4883  ‘cfv 6530  Topctop 22829  neicnei 23033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-top 22830  df-nei 23034

This theorem is referenced by:  hausflim  23917  flimclslem  23920  fclsfnflim  23963