Theorem ndfatafv2nrn 45915

Description: The alternate function value at a class 𝐴 at which the function is not defined is undefined, i.e., not in the range of the function. (Contributed by AV, 2-Sep-2022.)

Assertion

Ref	Expression
ndfatafv2nrn	⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) ∉ ran 𝐹)

Proof of Theorem ndfatafv2nrn

Step	Hyp	Ref	Expression
1		ndfatafv2 45905	. 2 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ∪ ran 𝐹)
2		pwuninel 8256	. . 3 ⊢ ¬ 𝒫 ∪ ran 𝐹 ∈ ran 𝐹
3		df-nel 3047	. . . 4 ⊢ ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ (𝐹''''𝐴) ∈ ran 𝐹)
4		eleq1 2821	. . . . 5 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → ((𝐹''''𝐴) ∈ ran 𝐹 ↔ 𝒫 ∪ ran 𝐹 ∈ ran 𝐹))
5	4	notbid 317	. . . 4 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → (¬ (𝐹''''𝐴) ∈ ran 𝐹 ↔ ¬ 𝒫 ∪ ran 𝐹 ∈ ran 𝐹))
6	3, 5	bitrid 282	. . 3 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ 𝒫 ∪ ran 𝐹 ∈ ran 𝐹))
7	2, 6	mpbiri 257	. 2 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → (𝐹''''𝐴) ∉ ran 𝐹)
8	1, 7	syl 17	1 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) ∉ ran 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1541   ∈ wcel 2106   ∉ wnel 3046  𝒫 cpw 4601  ∪ cuni 4907  ran crn 5676   defAt wdfat 45810  ''''cafv2 45902

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nel 3047  df-rab 3433  df-v 3476  df-un 3952  df-in 3954  df-ss 3964  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-uni 4908  df-afv2 45903

This theorem is referenced by:  ndmafv2nrn  45916  nfunsnafv2  45919  dfatafv2rnb  45921  tz6.12-2-afv2  45931