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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 6700. (Contributed by AV, 2-Sep-2022.) |
Ref | Expression |
---|---|
nfunsnafv2 | ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹''''𝐴) ∉ ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | olc 864 | . . 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 43309 | . . . 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 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → ¬ 𝐹 defAt 𝐴) |
6 | ndfatafv2nrn 43411 | . 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 398 ∨ wo 843 ∈ wcel 2108 ∉ wnel 3121 {csn 4559 dom cdm 5548 ran crn 5549 ↾ cres 5550 Fun wfun 6342 defAt wdfat 43306 ''''cafv2 43398 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7453 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-nel 3122 df-rex 3142 df-rab 3145 df-v 3495 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-uni 4831 df-dfat 43309 df-afv2 43399 |
This theorem is referenced by: (None) |
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