Theorem undefval 7289

Description: Value of the undefined value function. Normally we will not reference the explicit value but will use undefnel 7291 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		elex 3185	. 2 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V)
2		uniexg 6853	. . 3 ⊢ (𝑆 ∈ 𝑉 → ∪ 𝑆 ∈ V)
3		pwexg 4776	. . 3 ⊢ (∪ 𝑆 ∈ V → 𝒫 ∪ 𝑆 ∈ V)
4	2, 3	syl 17	. 2 ⊢ (𝑆 ∈ 𝑉 → 𝒫 ∪ 𝑆 ∈ V)
5		unieq 4380	. . . 4 ⊢ (𝑠 = 𝑆 → ∪ 𝑠 = ∪ 𝑆)
6	5	pweqd 4113	. . 3 ⊢ (𝑠 = 𝑆 → 𝒫 ∪ 𝑠 = 𝒫 ∪ 𝑆)
7		df-undef 7286	. . 3 ⊢ Undef = (𝑠 ∈ V ↦ 𝒫 ∪ 𝑠)
8	6, 7	fvmptg 6189	. 2 ⊢ ((𝑆 ∈ V ∧ 𝒫 ∪ 𝑆 ∈ V) → (Undef‘𝑆) = 𝒫 ∪ 𝑆)
9	1, 4, 8	syl2anc 691	1 ⊢ (𝑆 ∈ 𝑉 → (Undef‘𝑆) = 𝒫 ∪ 𝑆)

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  𝒫 cpw 4108  ∪ cuni 4372  ‘cfv 5804  Undefcund 7285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-undef 7286

This theorem is referenced by:  undefnel2  7290  undefne0  7292