Theorem neiuni 23087

Description: The union of the neighborhoods of a set equals the topology's underlying set. (Contributed by FL, 18-Sep-2007.) (Revised by Mario Carneiro, 9-Apr-2015.)

Hypothesis

Ref	Expression
tpnei.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
neiuni	⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑋 = ∪ ((nei‘𝐽)‘𝑆))

Proof of Theorem neiuni

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tpnei.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
2	1	tpnei 23086	. . . 4 ⊢ (𝐽 ∈ Top → (𝑆 ⊆ 𝑋 ↔ 𝑋 ∈ ((nei‘𝐽)‘𝑆)))
3	2	biimpa 476	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑋 ∈ ((nei‘𝐽)‘𝑆))
4		elssuni 4881	. . 3 ⊢ (𝑋 ∈ ((nei‘𝐽)‘𝑆) → 𝑋 ⊆ ∪ ((nei‘𝐽)‘𝑆))
5	3, 4	syl 17	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑋 ⊆ ∪ ((nei‘𝐽)‘𝑆))
6	1	neii1 23071	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆)) → 𝑥 ⊆ 𝑋)
7	6	ex 412	. . . . 5 ⊢ (𝐽 ∈ Top → (𝑥 ∈ ((nei‘𝐽)‘𝑆) → 𝑥 ⊆ 𝑋))
8	7	adantr 480	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ ((nei‘𝐽)‘𝑆) → 𝑥 ⊆ 𝑋))
9	8	ralrimiv 3128	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∀𝑥 ∈ ((nei‘𝐽)‘𝑆)𝑥 ⊆ 𝑋)
10		unissb 4883	. . 3 ⊢ (∪ ((nei‘𝐽)‘𝑆) ⊆ 𝑋 ↔ ∀𝑥 ∈ ((nei‘𝐽)‘𝑆)𝑥 ⊆ 𝑋)
11	9, 10	sylibr 234	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∪ ((nei‘𝐽)‘𝑆) ⊆ 𝑋)
12	5, 11	eqssd 3939	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑋 = ∪ ((nei‘𝐽)‘𝑆))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051   ⊆ wss 3889  ∪ cuni 4850  ‘cfv 6498  Topctop 22858  neicnei 23062

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-top 22859  df-nei 23063

This theorem is referenced by:  neifil  23845