Theorem ndfatafv2nrn 47958

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 47948	. 2 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ∪ ran 𝐹)
2		pwuninel 8267	. . 3 ⊢ ¬ 𝒫 ∪ ran 𝐹 ∈ ran 𝐹
3		df-nel 3065	. . . 4 ⊢ ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ (𝐹''''𝐴) ∈ ran 𝐹)
4		eleq1 2851	. . . . 5 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → ((𝐹''''𝐴) ∈ ran 𝐹 ↔ 𝒫 ∪ ran 𝐹 ∈ ran 𝐹))
5	4	notbid 321	. . . 4 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → (¬ (𝐹''''𝐴) ∈ ran 𝐹 ↔ ¬ 𝒫 ∪ ran 𝐹 ∈ ran 𝐹))
6	3, 5	bitrid 286	. . 3 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ 𝒫 ∪ ran 𝐹 ∈ ran 𝐹))
7	2, 6	mpbiri 261	. 2 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → (𝐹''''𝐴) ∉ ran 𝐹)
8	1, 7	syl 18	1 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) ∉ ran 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2143   ∉ wnel 3064  𝒫 cpw 4562  ∪ cuni 4872  ran crn 5662   defAt wdfat 47853  ''''cafv2 47945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nel 3065  df-rab 3417  df-v 3457  df-in 3912  df-ss 3922  df-if 4488  df-pw 4564  df-uni 4873  df-afv2 47946

This theorem is referenced by:  ndmafv2nrn  47959  nfunsnafv2  47962  dfatafv2rnb  47964  tz6.12-2-afv2  47974