Theorem undefval 7559

Description: Value of the undefined value function. Normally we will not reference the explicit value but will use undefnel 7561 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 3340	. 2 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V)
2		uniexg 7108	. . 3 ⊢ (𝑆 ∈ 𝑉 → ∪ 𝑆 ∈ V)
3		pwexg 4987	. . 3 ⊢ (∪ 𝑆 ∈ V → 𝒫 ∪ 𝑆 ∈ V)
4	2, 3	syl 17	. 2 ⊢ (𝑆 ∈ 𝑉 → 𝒫 ∪ 𝑆 ∈ V)
5		unieq 4584	. . . 4 ⊢ (𝑠 = 𝑆 → ∪ 𝑠 = ∪ 𝑆)
6	5	pweqd 4295	. . 3 ⊢ (𝑠 = 𝑆 → 𝒫 ∪ 𝑠 = 𝒫 ∪ 𝑆)
7		df-undef 7556	. . 3 ⊢ Undef = (𝑠 ∈ V ↦ 𝒫 ∪ 𝑠)
8	6, 7	fvmptg 6430	. 2 ⊢ ((𝑆 ∈ V ∧ 𝒫 ∪ 𝑆 ∈ V) → (Undef‘𝑆) = 𝒫 ∪ 𝑆)
9	1, 4, 8	syl2anc 696	1 ⊢ (𝑆 ∈ 𝑉 → (Undef‘𝑆) = 𝒫 ∪ 𝑆)

Syntax hints:   → wi 4   = wceq 1620   ∈ wcel 2127  Vcvv 3328  𝒫 cpw 4290  ∪ cuni 4576  ‘cfv 6037  Undefcund 7555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-iota 6000  df-fun 6039  df-fv 6045  df-undef 7556

This theorem is referenced by:  undefnel2  7560  undefne0  7562