Theorem neiss2 13727

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 13726	. . . . 5 ⊢ (𝐽 ∈ Top → (nei‘𝐽) Fn 𝒫 𝑋)
3		fnrel 5316	. . . . 5 ⊢ ((nei‘𝐽) Fn 𝒫 𝑋 → Rel (nei‘𝐽))
4	2, 3	syl 14	. . . 4 ⊢ (𝐽 ∈ Top → Rel (nei‘𝐽))
5		relelfvdm 5549	. . . 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 5317	. . . . . 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 2247	. . . 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 3588	1 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ⊆ 𝑋)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148   ⊆ wss 3131  𝒫 cpw 3577  ∪ cuni 3811  dom cdm 4628  Rel wrel 4633   Fn wfn 5213  ‘cfv 5218  Topctop 13582  neicnei 13723

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-top 13583  df-nei 13724

This theorem is referenced by:  neii1  13732  neii2  13734  neiss  13735  ssnei2  13742  topssnei  13747  innei  13748  neitx  13853