Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ddeval1 Structured version   Visualization version   GIF version

Theorem ddeval1 31392
Description: Value of the delta measure. (Contributed by Thierry Arnoux, 14-Sep-2018.)
Assertion
Ref Expression
ddeval1 ((𝐴 ⊆ ℝ ∧ 0 ∈ 𝐴) → (δ‘𝐴) = 1)

Proof of Theorem ddeval1
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 reex 10616 . . . . 5 ℝ ∈ V
21ssex 5216 . . . 4 (𝐴 ⊆ ℝ → 𝐴 ∈ V)
3 elpwg 4541 . . . . 5 (𝐴 ∈ V → (𝐴 ∈ 𝒫 ℝ ↔ 𝐴 ⊆ ℝ))
43biimpar 478 . . . 4 ((𝐴 ∈ V ∧ 𝐴 ⊆ ℝ) → 𝐴 ∈ 𝒫 ℝ)
52, 4mpancom 684 . . 3 (𝐴 ⊆ ℝ → 𝐴 ∈ 𝒫 ℝ)
6 eleq2 2898 . . . . 5 (𝑎 = 𝐴 → (0 ∈ 𝑎 ↔ 0 ∈ 𝐴))
76ifbid 4485 . . . 4 (𝑎 = 𝐴 → if(0 ∈ 𝑎, 1, 0) = if(0 ∈ 𝐴, 1, 0))
8 df-dde 31391 . . . 4 δ = (𝑎 ∈ 𝒫 ℝ ↦ if(0 ∈ 𝑎, 1, 0))
9 1ex 10625 . . . . 5 1 ∈ V
10 c0ex 10623 . . . . 5 0 ∈ V
119, 10ifex 4511 . . . 4 if(0 ∈ 𝐴, 1, 0) ∈ V
127, 8, 11fvmpt 6761 . . 3 (𝐴 ∈ 𝒫 ℝ → (δ‘𝐴) = if(0 ∈ 𝐴, 1, 0))
135, 12syl 17 . 2 (𝐴 ⊆ ℝ → (δ‘𝐴) = if(0 ∈ 𝐴, 1, 0))
14 iftrue 4469 . 2 (0 ∈ 𝐴 → if(0 ∈ 𝐴, 1, 0) = 1)
1513, 14sylan9eq 2873 1 ((𝐴 ⊆ ℝ ∧ 0 ∈ 𝐴) → (δ‘𝐴) = 1)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  wss 3933  ifcif 4463  𝒫 cpw 4535  cfv 6348  cr 10524  0cc0 10525  1c1 10526  δcdde 31390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-mulcl 10587  ax-i2m1 10593
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-dde 31391
This theorem is referenced by:  ddemeas  31394
  Copyright terms: Public domain W3C validator