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Theorem undefval 7684
Description: Value of the undefined value function. Normally we will not reference the explicit value but will use undefnel 7686 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 7681 . 2 Undef = (𝑠 ∈ V ↦ 𝒫 𝑠)
2 unieq 4679 . . 3 (𝑠 = 𝑆 𝑠 = 𝑆)
32pweqd 4384 . 2 (𝑠 = 𝑆 → 𝒫 𝑠 = 𝒫 𝑆)
4 elex 3414 . 2 (𝑆𝑉𝑆 ∈ V)
5 uniexg 7232 . . 3 (𝑆𝑉 𝑆 ∈ V)
65pwexd 5091 . 2 (𝑆𝑉 → 𝒫 𝑆 ∈ V)
71, 3, 4, 6fvmptd3 6564 1 (𝑆𝑉 → (Undef‘𝑆) = 𝒫 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wcel 2107  Vcvv 3398  𝒫 cpw 4379   cuni 4671  cfv 6135  Undefcund 7680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fv 6143  df-undef 7681
This theorem is referenced by:  undefnel2  7685  undefne0  7687  ndfatafv2undef  42253
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