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Theorem ddeval1 30627
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 10239 . . . . 5 ℝ ∈ V
21ssex 4954 . . . 4 (𝐴 ⊆ ℝ → 𝐴 ∈ V)
3 elpwg 4310 . . . . 5 (𝐴 ∈ V → (𝐴 ∈ 𝒫 ℝ ↔ 𝐴 ⊆ ℝ))
43biimpar 503 . . . 4 ((𝐴 ∈ V ∧ 𝐴 ⊆ ℝ) → 𝐴 ∈ 𝒫 ℝ)
52, 4mpancom 706 . . 3 (𝐴 ⊆ ℝ → 𝐴 ∈ 𝒫 ℝ)
6 eleq2 2828 . . . . 5 (𝑎 = 𝐴 → (0 ∈ 𝑎 ↔ 0 ∈ 𝐴))
76ifbid 4252 . . . 4 (𝑎 = 𝐴 → if(0 ∈ 𝑎, 1, 0) = if(0 ∈ 𝐴, 1, 0))
8 df-dde 30626 . . . 4 δ = (𝑎 ∈ 𝒫 ℝ ↦ if(0 ∈ 𝑎, 1, 0))
9 1ex 10247 . . . . 5 1 ∈ V
10 c0ex 10246 . . . . 5 0 ∈ V
119, 10ifex 4300 . . . 4 if(0 ∈ 𝐴, 1, 0) ∈ V
127, 8, 11fvmpt 6445 . . 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 4236 . 2 (0 ∈ 𝐴 → if(0 ∈ 𝐴, 1, 0) = 1)
1513, 14sylan9eq 2814 1 ((𝐴 ⊆ ℝ ∧ 0 ∈ 𝐴) → (δ‘𝐴) = 1)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  wss 3715  ifcif 4230  𝒫 cpw 4302  cfv 6049  cr 10147  0cc0 10148  1c1 10149  δcdde 30625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-mulcl 10210  ax-i2m1 10216
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-dde 30626
This theorem is referenced by:  ddemeas  30629
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