Theorem neiuni 23009

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 23008	. . . 4 ⊢ (𝐽 ∈ Top → (𝑆 ⊆ 𝑋 ↔ 𝑋 ∈ ((nei‘𝐽)‘𝑆)))
3	2	biimpa 476	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑋 ∈ ((nei‘𝐽)‘𝑆))
4		elssuni 4901	. . 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 22993	. . . . . 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 3124	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∀𝑥 ∈ ((nei‘𝐽)‘𝑆)𝑥 ⊆ 𝑋)
10		unissb 4903	. . 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 3964	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑋 = ∪ ((nei‘𝐽)‘𝑆))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ⊆ wss 3914  ∪ cuni 4871  ‘cfv 6511  Topctop 22780  neicnei 22984

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-top 22781  df-nei 22985

This theorem is referenced by:  neifil  23767