Theorem ndfatafv2nrn 46514

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 46504	. 2 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ∪ ran 𝐹)
2		pwuninel 8272	. . 3 ⊢ ¬ 𝒫 ∪ ran 𝐹 ∈ ran 𝐹
3		df-nel 3042	. . . 4 ⊢ ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ (𝐹''''𝐴) ∈ ran 𝐹)
4		eleq1 2816	. . . . 5 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → ((𝐹''''𝐴) ∈ ran 𝐹 ↔ 𝒫 ∪ ran 𝐹 ∈ ran 𝐹))
5	4	notbid 318	. . . 4 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → (¬ (𝐹''''𝐴) ∈ ran 𝐹 ↔ ¬ 𝒫 ∪ ran 𝐹 ∈ ran 𝐹))
6	3, 5	bitrid 283	. . 3 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ 𝒫 ∪ ran 𝐹 ∈ ran 𝐹))
7	2, 6	mpbiri 258	. 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 1534   ∈ wcel 2099   ∉ wnel 3041  𝒫 cpw 4598  ∪ cuni 4903  ran crn 5673   defAt wdfat 46409  ''''cafv2 46501

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-sep 5293  ax-pr 5423  ax-un 7732

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-nel 3042  df-rab 3428  df-v 3471  df-un 3949  df-in 3951  df-ss 3961  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-uni 4904  df-afv2 46502

This theorem is referenced by:  ndmafv2nrn  46515  nfunsnafv2  46518  dfatafv2rnb  46520  tz6.12-2-afv2  46530