MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  neiss2 Structured version   Visualization version   GIF version

Theorem neiss2 22358
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 = βˆͺ 𝐽
Assertion
Ref Expression
neiss2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ 𝑆 βŠ† 𝑋)

Proof of Theorem neiss2
StepHypRef Expression
1 elfvdm 6862 . . . 4 (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) β†’ 𝑆 ∈ dom (neiβ€˜π½))
21adantl 482 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ 𝑆 ∈ dom (neiβ€˜π½))
3 neifval.1 . . . . . . 7 𝑋 = βˆͺ 𝐽
43neif 22357 . . . . . 6 (𝐽 ∈ Top β†’ (neiβ€˜π½) Fn 𝒫 𝑋)
54fndmd 6590 . . . . 5 (𝐽 ∈ Top β†’ dom (neiβ€˜π½) = 𝒫 𝑋)
65eleq2d 2822 . . . 4 (𝐽 ∈ Top β†’ (𝑆 ∈ dom (neiβ€˜π½) ↔ 𝑆 ∈ 𝒫 𝑋))
76adantr 481 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ (𝑆 ∈ dom (neiβ€˜π½) ↔ 𝑆 ∈ 𝒫 𝑋))
82, 7mpbid 231 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ 𝑆 ∈ 𝒫 𝑋)
98elpwid 4556 1 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ 𝑆 βŠ† 𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1540   ∈ wcel 2105   βŠ† wss 3898  π’« cpw 4547  βˆͺ cuni 4852  dom cdm 5620  β€˜cfv 6479  Topctop 22148  neicnei 22354
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-top 22149  df-nei 22355
This theorem is referenced by:  neii1  22363  neii2  22365  neiss  22366  ssnei2  22373  topssnei  22381  innei  22382  neitx  22864  cvmlift2lem12  33575  neiin  34617  cnneiima  46561
  Copyright terms: Public domain W3C validator