Theorem ndmafv2nrn 43414

Description: The value of a class outside its domain is not in the range, compare with ndmfv 6695. (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 863	. . 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 43311	. . . 4 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
4	2, 3	xchnxbir 335	. . 3 ⊢ (¬ 𝐹 defAt 𝐴 ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
5	1, 4	sylibr 236	. 2 ⊢ (¬ 𝐴 ∈ dom 𝐹 → ¬ 𝐹 defAt 𝐴)
6		ndfatafv2nrn 43413	. 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 398   ∨ wo 843   ∈ wcel 2110   ∉ wnel 3123  {csn 4561  dom cdm 5550  ran crn 5551   ↾ cres 5552  Fun wfun 6344   defAt wdfat 43308  ''''cafv2 43400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-nel 3124  df-rex 3144  df-rab 3147  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-uni 4833  df-dfat 43311  df-afv2 43401

This theorem is referenced by:  afv2prc  43418  fafv2elrnb  43427