Theorem undefval 8301

Description: Value of the undefined value function. Normally we will not reference the explicit value but will use undefnel 8303 instead. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 24-Dec-2016.)

Assertion

Ref	Expression
undefval	⊢ (𝑆 ∈ 𝑉 → (Undef‘𝑆) = 𝒫 ∪ 𝑆)

Proof of Theorem undefval

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-undef 8298	. 2 ⊢ Undef = (𝑠 ∈ V ↦ 𝒫 ∪ 𝑠)
2		unieq 4918	. . 3 ⊢ (𝑠 = 𝑆 → ∪ 𝑠 = ∪ 𝑆)
3	2	pweqd 4617	. 2 ⊢ (𝑠 = 𝑆 → 𝒫 ∪ 𝑠 = 𝒫 ∪ 𝑆)
4		elex 3501	. 2 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V)
5		uniexg 7760	. . 3 ⊢ (𝑆 ∈ 𝑉 → ∪ 𝑆 ∈ V)
6	5	pwexd 5379	. 2 ⊢ (𝑆 ∈ 𝑉 → 𝒫 ∪ 𝑆 ∈ V)
7	1, 3, 4, 6	fvmptd3 7039	1 ⊢ (𝑆 ∈ 𝑉 → (Undef‘𝑆) = 𝒫 ∪ 𝑆)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  Vcvv 3480  𝒫 cpw 4600  ∪ cuni 4907  ‘cfv 6561  Undefcund 8297

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-undef 8298

This theorem is referenced by:  undefnel2  8302  undefne0  8304  ndfatafv2undef  47224