Theorem ssnei2 23029

Description: Any subset 𝑀 of 𝑋 containing a neighborhood 𝑁 of a set 𝑆 is a neighborhood of this set. Generalization to subsets of Property V_i of [BourbakiTop1] p. I.3. (Contributed by FL, 2-Oct-2006.)

Hypothesis

Ref	Expression
neips.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
ssnei2	⊢ (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑁 ⊆ 𝑀 ∧ 𝑀 ⊆ 𝑋)) → 𝑀 ∈ ((nei‘𝐽)‘𝑆))

Proof of Theorem ssnei2

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprr 772	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑁 ⊆ 𝑀 ∧ 𝑀 ⊆ 𝑋)) → 𝑀 ⊆ 𝑋)
2		neii2 23021	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁))
3		sstr2 3941	. . . . . . 7 ⊢ (𝑔 ⊆ 𝑁 → (𝑁 ⊆ 𝑀 → 𝑔 ⊆ 𝑀))
4	3	com12 32	. . . . . 6 ⊢ (𝑁 ⊆ 𝑀 → (𝑔 ⊆ 𝑁 → 𝑔 ⊆ 𝑀))
5	4	anim2d 612	. . . . 5 ⊢ (𝑁 ⊆ 𝑀 → ((𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁) → (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑀)))
6	5	reximdv 3147	. . . 4 ⊢ (𝑁 ⊆ 𝑀 → (∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑀)))
7	2, 6	mpan9 506	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑁 ⊆ 𝑀) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑀))
8	7	adantrr 717	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑁 ⊆ 𝑀 ∧ 𝑀 ⊆ 𝑋)) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑀))
9		neips.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
10	9	neiss2 23014	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ⊆ 𝑋)
11	9	isnei 23016	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑀 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑀 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑀))))
12	10, 11	syldan 591	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → (𝑀 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑀 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑀))))
13	12	adantr 480	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑁 ⊆ 𝑀 ∧ 𝑀 ⊆ 𝑋)) → (𝑀 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑀 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑀))))
14	1, 8, 13	mpbir2and 713	1 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑁 ⊆ 𝑀 ∧ 𝑀 ⊆ 𝑋)) → 𝑀 ∈ ((nei‘𝐽)‘𝑆))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∃wrex 3056   ⊆ wss 3902  ∪ cuni 4859  ‘cfv 6481  Topctop 22806  neicnei 23010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-top 22807  df-nei 23011

This theorem is referenced by:  topssnei  23037  nllyrest  23399  nllyidm  23402  hausllycmp  23407  cldllycmp  23408  txnlly  23550  neifil  23793  utop2nei  24163  cnllycmp  24880  gneispb  44163