Theorem ndmafv2nrn 46507

Description: The value of a class outside its domain is not in the range, compare with ndmfv 6920. (Contributed by AV, 2-Sep-2022.)

Assertion

Ref	Expression
ndmafv2nrn	⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹''''𝐴) ∉ ran 𝐹)

Proof of Theorem ndmafv2nrn

Step	Hyp	Ref	Expression
1		orc 864	. . 3 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
2		ianor 978	. . . 4 ⊢ (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
3		df-dfat 46404	. . . 4 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
4	2, 3	xchnxbir 333	. . 3 ⊢ (¬ 𝐹 defAt 𝐴 ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
5	1, 4	sylibr 233	. 2 ⊢ (¬ 𝐴 ∈ dom 𝐹 → ¬ 𝐹 defAt 𝐴)
6		ndfatafv2nrn 46506	. 2 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) ∉ ran 𝐹)
7	5, 6	syl 17	1 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹''''𝐴) ∉ ran 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 844   ∈ wcel 2098   ∉ wnel 3040  {csn 4623  dom cdm 5669  ran crn 5670   ↾ cres 5671  Fun wfun 6531   defAt wdfat 46401  ''''cafv2 46493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-pr 5420  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nel 3041  df-rab 3427  df-v 3470  df-un 3948  df-in 3950  df-ss 3960  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-uni 4903  df-dfat 46404  df-afv2 46494

This theorem is referenced by:  afv2prc  46511  fafv2elrnb  46520