Theorem undefval 8279

Description: Value of the undefined value function. Normally we will not reference the explicit value but will use undefnel 8281 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 8275	. 2 ⊢ Undef = (𝑠 ∈ V ↦ 𝒫 ∪ 𝑠)
2		unieq 4881	. . 3 ⊢ (𝑠 = 𝑆 → ∪ 𝑠 = ∪ 𝑆)
3	2	pweqd 4577	. 2 ⊢ (𝑠 = 𝑆 → 𝒫 ∪ 𝑠 = 𝒫 ∪ 𝑆)
4		elex 3474	. 2 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V)
5		uniexg 7746	. . 3 ⊢ (𝑆 ∈ 𝑉 → ∪ 𝑆 ∈ V)
6	5	pwexd 5348	. 2 ⊢ (𝑆 ∈ 𝑉 → 𝒫 ∪ 𝑆 ∈ V)
7	1, 3, 4, 6	fvmptd3 7014	1 ⊢ (𝑆 ∈ 𝑉 → (Undef‘𝑆) = 𝒫 ∪ 𝑆)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  Vcvv 3453  𝒫 cpw 4560  ∪ cuni 4870  ‘cfv 6537  Undefcund 8274

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6493  df-fun 6539  df-fv 6545  df-undef 8275

This theorem is used by:  undefnel2  8280  undefne0  8282  ndfatafv2undef  48108