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Theorem undefval 8300
Description: Value of the undefined value function. Normally we will not reference the explicit value but will use undefnel 8302 instead. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 24-Dec-2016.)
Assertion
Ref Expression
undefval (𝑆𝑉 → (Undef‘𝑆) = 𝒫 𝑆)

Proof of Theorem undefval
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 df-undef 8297 . 2 Undef = (𝑠 ∈ V ↦ 𝒫 𝑠)
2 unieq 4923 . . 3 (𝑠 = 𝑆 𝑠 = 𝑆)
32pweqd 4622 . 2 (𝑠 = 𝑆 → 𝒫 𝑠 = 𝒫 𝑆)
4 elex 3499 . 2 (𝑆𝑉𝑆 ∈ V)
5 uniexg 7759 . . 3 (𝑆𝑉 𝑆 ∈ V)
65pwexd 5385 . 2 (𝑆𝑉 → 𝒫 𝑆 ∈ V)
71, 3, 4, 6fvmptd3 7039 1 (𝑆𝑉 → (Undef‘𝑆) = 𝒫 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  Vcvv 3478  𝒫 cpw 4605   cuni 4912  cfv 6563  Undefcund 8296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-undef 8297
This theorem is referenced by:  undefnel2  8301  undefne0  8303  ndfatafv2undef  47162
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