Theorem undefval 7944

Description: Value of the undefined value function. Normally we will not reference the explicit value but will use undefnel 7946 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 7941	. 2 ⊢ Undef = (𝑠 ∈ V ↦ 𝒫 ∪ 𝑠)
2		unieq 4851	. . 3 ⊢ (𝑠 = 𝑆 → ∪ 𝑠 = ∪ 𝑆)
3	2	pweqd 4560	. 2 ⊢ (𝑠 = 𝑆 → 𝒫 ∪ 𝑠 = 𝒫 ∪ 𝑆)
4		elex 3514	. 2 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V)
5		uniexg 7468	. . 3 ⊢ (𝑆 ∈ 𝑉 → ∪ 𝑆 ∈ V)
6	5	pwexd 5282	. 2 ⊢ (𝑆 ∈ 𝑉 → 𝒫 ∪ 𝑆 ∈ V)
7	1, 3, 4, 6	fvmptd3 6793	1 ⊢ (𝑆 ∈ 𝑉 → (Undef‘𝑆) = 𝒫 ∪ 𝑆)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  Vcvv 3496  𝒫 cpw 4541  ∪ cuni 4840  ‘cfv 6357  Undefcund 7940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-iota 6316  df-fun 6359  df-fv 6365  df-undef 7941

This theorem is referenced by:  undefnel2  7945  undefne0  7947  ndfatafv2undef  43418