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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > nfunsnafv | Structured version Visualization version GIF version |
Description: If the restriction of a class to a singleton is not a function, its value is the universe, compare with nfunsn 6135. (Contributed by Alexander van der Vekens, 25-May-2017.) |
Ref | Expression |
---|---|
nfunsnafv | ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹'''𝐴) = V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dfat 39845 | . . . 4 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | |
2 | 1 | simprbi 479 | . . 3 ⊢ (𝐹 defAt 𝐴 → Fun (𝐹 ↾ {𝐴})) |
3 | 2 | con3i 149 | . 2 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → ¬ 𝐹 defAt 𝐴) |
4 | afvnfundmuv 39868 | . 2 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹'''𝐴) = V) | |
5 | 3, 4 | syl 17 | 1 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹'''𝐴) = V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 {csn 4125 dom cdm 5038 ↾ cres 5040 Fun wfun 5798 defAt wdfat 39842 '''cafv 39843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-un 3545 df-if 4037 df-fv 5812 df-dfat 39845 df-afv 39846 |
This theorem is referenced by: afvvfunressn 39872 nfunsnaov 39915 |
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