Theorem tpnei 21254

Description: The underlying set of a topology is a neighborhood of any of its subsets. Special case of opnneiss 21251. (Contributed by FL, 2-Oct-2006.)

Hypothesis

Ref	Expression
tpnei.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
tpnei	⊢ (𝐽 ∈ Top → (𝑆 ⊆ 𝑋 ↔ 𝑋 ∈ ((nei‘𝐽)‘𝑆)))

Proof of Theorem tpnei

Step	Hyp	Ref	Expression
1		tpnei.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
2	1	topopn 21039	. . 3 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
3		opnneiss 21251	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑋) → 𝑋 ∈ ((nei‘𝐽)‘𝑆))
4	3	3exp 1149	. . 3 ⊢ (𝐽 ∈ Top → (𝑋 ∈ 𝐽 → (𝑆 ⊆ 𝑋 → 𝑋 ∈ ((nei‘𝐽)‘𝑆))))
5	2, 4	mpd 15	. 2 ⊢ (𝐽 ∈ Top → (𝑆 ⊆ 𝑋 → 𝑋 ∈ ((nei‘𝐽)‘𝑆)))
6		ssnei 21243	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ⊆ 𝑋)
7	6	ex 402	. 2 ⊢ (𝐽 ∈ Top → (𝑋 ∈ ((nei‘𝐽)‘𝑆) → 𝑆 ⊆ 𝑋))
8	5, 7	impbid 204	1 ⊢ (𝐽 ∈ Top → (𝑆 ⊆ 𝑋 ↔ 𝑋 ∈ ((nei‘𝐽)‘𝑆)))

Syntax hints:   → wi 4   ↔ wb 198   = wceq 1653   ∈ wcel 2157   ⊆ wss 3769  ∪ cuni 4628  ‘cfv 6101  Topctop 21026  neicnei 21230

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-top 21027  df-nei 21231

This theorem is referenced by:  neiuni  21255  neifil  22012  gneispa  39210