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Theorem ddeval1 33762
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 11200 . . . . 5 ℝ ∈ V
21ssex 5314 . . . 4 (𝐴 ⊆ ℝ → 𝐴 ∈ V)
3 elpwg 4600 . . . . 5 (𝐴 ∈ V → (𝐴 ∈ 𝒫 ℝ ↔ 𝐴 ⊆ ℝ))
43biimpar 477 . . . 4 ((𝐴 ∈ V ∧ 𝐴 ⊆ ℝ) → 𝐴 ∈ 𝒫 ℝ)
52, 4mpancom 685 . . 3 (𝐴 ⊆ ℝ → 𝐴 ∈ 𝒫 ℝ)
6 eleq2 2816 . . . . 5 (𝑎 = 𝐴 → (0 ∈ 𝑎 ↔ 0 ∈ 𝐴))
76ifbid 4546 . . . 4 (𝑎 = 𝐴 → if(0 ∈ 𝑎, 1, 0) = if(0 ∈ 𝐴, 1, 0))
8 df-dde 33761 . . . 4 δ = (𝑎 ∈ 𝒫 ℝ ↦ if(0 ∈ 𝑎, 1, 0))
9 1ex 11211 . . . . 5 1 ∈ V
10 c0ex 11209 . . . . 5 0 ∈ V
119, 10ifex 4573 . . . 4 if(0 ∈ 𝐴, 1, 0) ∈ V
127, 8, 11fvmpt 6991 . . 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 4529 . 2 (0 ∈ 𝐴 → if(0 ∈ 𝐴, 1, 0) = 1)
1513, 14sylan9eq 2786 1 ((𝐴 ⊆ ℝ ∧ 0 ∈ 𝐴) → (δ‘𝐴) = 1)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  Vcvv 3468  wss 3943  ifcif 4523  𝒫 cpw 4597  cfv 6536  cr 11108  0cc0 11109  1c1 11110  δcdde 33760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-mulcl 11171  ax-i2m1 11177
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-dde 33761
This theorem is referenced by:  ddemeas  33764
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