Theorem undefval 7402

Description: Value of the undefined value function. Normally we will not reference the explicit value but will use undefnel 7404 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 3212	. 2 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V)
2		uniexg 6955	. . 3 ⊢ (𝑆 ∈ 𝑉 → ∪ 𝑆 ∈ V)
3		pwexg 4850	. . 3 ⊢ (∪ 𝑆 ∈ V → 𝒫 ∪ 𝑆 ∈ V)
4	2, 3	syl 17	. 2 ⊢ (𝑆 ∈ 𝑉 → 𝒫 ∪ 𝑆 ∈ V)
5		unieq 4444	. . . 4 ⊢ (𝑠 = 𝑆 → ∪ 𝑠 = ∪ 𝑆)
6	5	pweqd 4163	. . 3 ⊢ (𝑠 = 𝑆 → 𝒫 ∪ 𝑠 = 𝒫 ∪ 𝑆)
7		df-undef 7399	. . 3 ⊢ Undef = (𝑠 ∈ V ↦ 𝒫 ∪ 𝑠)
8	6, 7	fvmptg 6280	. 2 ⊢ ((𝑆 ∈ V ∧ 𝒫 ∪ 𝑆 ∈ V) → (Undef‘𝑆) = 𝒫 ∪ 𝑆)
9	1, 4, 8	syl2anc 693	1 ⊢ (𝑆 ∈ 𝑉 → (Undef‘𝑆) = 𝒫 ∪ 𝑆)

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  Vcvv 3200  𝒫 cpw 4158  ∪ cuni 4436  ‘cfv 5888  Undefcund 7398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-undef 7399

This theorem is referenced by:  undefnel2  7403  undefne0  7405