Theorem neiuni 21414

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 21413	. . . 4 ⊢ (𝐽 ∈ Top → (𝑆 ⊆ 𝑋 ↔ 𝑋 ∈ ((nei‘𝐽)‘𝑆)))
3	2	biimpa 477	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑋 ∈ ((nei‘𝐽)‘𝑆))
4		elssuni 4774	. . 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 21398	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆)) → 𝑥 ⊆ 𝑋)
7	6	ex 413	. . . . 5 ⊢ (𝐽 ∈ Top → (𝑥 ∈ ((nei‘𝐽)‘𝑆) → 𝑥 ⊆ 𝑋))
8	7	adantr 481	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ ((nei‘𝐽)‘𝑆) → 𝑥 ⊆ 𝑋))
9	8	ralrimiv 3148	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∀𝑥 ∈ ((nei‘𝐽)‘𝑆)𝑥 ⊆ 𝑋)
10		unissb 4776	. . 3 ⊢ (∪ ((nei‘𝐽)‘𝑆) ⊆ 𝑋 ↔ ∀𝑥 ∈ ((nei‘𝐽)‘𝑆)𝑥 ⊆ 𝑋)
11	9, 10	sylibr 235	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∪ ((nei‘𝐽)‘𝑆) ⊆ 𝑋)
12	5, 11	eqssd 3906	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑋 = ∪ ((nei‘𝐽)‘𝑆))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1522   ∈ wcel 2081  ∀wral 3105   ⊆ wss 3859  ∪ cuni 4745  ‘cfv 6225  Topctop 21185  neicnei 21389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-top 21186  df-nei 21390

This theorem is referenced by:  neifil  22172