![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > nfunsnafv2 | Structured version Visualization version GIF version |
Description: If the restriction of a class to a singleton is not a function, its value at the singleton element is undefined, compare with nfunsn 6927. (Contributed by AV, 2-Sep-2022.) |
Ref | Expression |
---|---|
nfunsnafv2 | ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹''''𝐴) ∉ ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | olc 865 | . . 3 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (¬ 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 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → ¬ 𝐹 defAt 𝐴) |
6 | ndfatafv2nrn 46506 | . 2 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) ∉ ran 𝐹) | |
7 | 5, 6 | syl 17 | 1 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹''''𝐴) ∉ ran 𝐹) |
Colors of variables: wff setvar class |
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: (None) |
Copyright terms: Public domain | W3C validator |