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Theorem undefval 8211
Description: Value of the undefined value function. Normally we will not reference the explicit value but will use undefnel 8213 instead. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 24-Dec-2016.)
Assertion
Ref Expression
undefval (𝑆 ∈ 𝑉 β†’ (Undefβ€˜π‘†) = 𝒫 βˆͺ 𝑆)

Proof of Theorem undefval
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 df-undef 8208 . 2 Undef = (𝑠 ∈ V ↦ 𝒫 βˆͺ 𝑠)
2 unieq 4880 . . 3 (𝑠 = 𝑆 β†’ βˆͺ 𝑠 = βˆͺ 𝑆)
32pweqd 4581 . 2 (𝑠 = 𝑆 β†’ 𝒫 βˆͺ 𝑠 = 𝒫 βˆͺ 𝑆)
4 elex 3465 . 2 (𝑆 ∈ 𝑉 β†’ 𝑆 ∈ V)
5 uniexg 7681 . . 3 (𝑆 ∈ 𝑉 β†’ βˆͺ 𝑆 ∈ V)
65pwexd 5338 . 2 (𝑆 ∈ 𝑉 β†’ 𝒫 βˆͺ 𝑆 ∈ V)
71, 3, 4, 6fvmptd3 6975 1 (𝑆 ∈ 𝑉 β†’ (Undefβ€˜π‘†) = 𝒫 βˆͺ 𝑆)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  Vcvv 3447  π’« cpw 4564  βˆͺ cuni 4869  β€˜cfv 6500  Undefcund 8207
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-undef 8208
This theorem is referenced by:  undefnel2  8212  undefne0  8214  ndfatafv2undef  45534
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