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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 6949. (Contributed by AV, 2-Sep-2022.) |
Ref | Expression |
---|---|
nfunsnafv2 | ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹''''𝐴) ∉ ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | olc 868 | . . 3 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴}))) | |
2 | ianor 983 | . . . 4 ⊢ (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴}))) | |
3 | df-dfat 47069 | . . . 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 234 | . 2 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → ¬ 𝐹 defAt 𝐴) |
6 | ndfatafv2nrn 47171 | . 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 847 ∈ wcel 2106 ∉ wnel 3044 {csn 4631 dom cdm 5689 ran crn 5690 ↾ cres 5691 Fun wfun 6557 defAt wdfat 47066 ''''cafv2 47158 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nel 3045 df-rab 3434 df-v 3480 df-un 3968 df-in 3970 df-ss 3980 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-uni 4913 df-dfat 47069 df-afv2 47159 |
This theorem is referenced by: (None) |
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