Theorem neiss2 14689

Description: A set with a neighborhood is a subset of the base set of a topology. (This theorem depends on a function's value being empty outside of its domain, but it will make later theorems simpler to state.) (Contributed by NM, 12-Feb-2007.)

Hypothesis

Ref	Expression
neifval.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
neiss2	⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ⊆ 𝑋)

Proof of Theorem neiss2

Step	Hyp	Ref	Expression
1		neifval.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
2	1	neif 14688	. . . . 5 ⊢ (𝐽 ∈ Top → (nei‘𝐽) Fn 𝒫 𝑋)
3		fnrel 5381	. . . . 5 ⊢ ((nei‘𝐽) Fn 𝒫 𝑋 → Rel (nei‘𝐽))
4	2, 3	syl 14	. . . 4 ⊢ (𝐽 ∈ Top → Rel (nei‘𝐽))
5		relelfvdm 5621	. . . 4 ⊢ ((Rel (nei‘𝐽) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ∈ dom (nei‘𝐽))
6	4, 5	sylan 283	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ∈ dom (nei‘𝐽))
7		fndm 5382	. . . . . 6 ⊢ ((nei‘𝐽) Fn 𝒫 𝑋 → dom (nei‘𝐽) = 𝒫 𝑋)
8	2, 7	syl 14	. . . . 5 ⊢ (𝐽 ∈ Top → dom (nei‘𝐽) = 𝒫 𝑋)
9	8	eleq2d 2276	. . . 4 ⊢ (𝐽 ∈ Top → (𝑆 ∈ dom (nei‘𝐽) ↔ 𝑆 ∈ 𝒫 𝑋))
10	9	adantr 276	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → (𝑆 ∈ dom (nei‘𝐽) ↔ 𝑆 ∈ 𝒫 𝑋))
11	6, 10	mpbid 147	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ∈ 𝒫 𝑋)
12	11	elpwid 3632	1 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ⊆ 𝑋)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1373   ∈ wcel 2177   ⊆ wss 3170  𝒫 cpw 3621  ∪ cuni 3856  dom cdm 4683  Rel wrel 4688   Fn wfn 5275  ‘cfv 5280  Topctop 14544  neicnei 14685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-pow 4226  ax-pr 4261

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-top 14545  df-nei 14686

This theorem is referenced by:  neii1  14694  neii2  14696  neiss  14697  ssnei2  14704  topssnei  14709  innei  14710  neitx  14815