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Theorem undefval 7934
Description: Value of the undefined value function. Normally we will not reference the explicit value but will use undefnel 7936 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 7931 . 2 Undef = (𝑠 ∈ V ↦ 𝒫 𝑠)
2 unieq 4836 . . 3 (𝑠 = 𝑆 𝑠 = 𝑆)
32pweqd 4541 . 2 (𝑠 = 𝑆 → 𝒫 𝑠 = 𝒫 𝑆)
4 elex 3498 . 2 (𝑆𝑉𝑆 ∈ V)
5 uniexg 7457 . . 3 (𝑆𝑉 𝑆 ∈ V)
65pwexd 5268 . 2 (𝑆𝑉 → 𝒫 𝑆 ∈ V)
71, 3, 4, 6fvmptd3 6780 1 (𝑆𝑉 → (Undef‘𝑆) = 𝒫 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  Vcvv 3480  𝒫 cpw 4522   cuni 4825  cfv 6344  Undefcund 7930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-iota 6303  df-fun 6346  df-fv 6352  df-undef 7931
This theorem is referenced by:  undefnel2  7935  undefne0  7937  ndfatafv2undef  43634
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