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Mirrors > Home > MPE Home > Th. List > Mathboxes > aovvfunressn | Structured version Visualization version GIF version |
Description: If the operation value of a class for an argument is a set, the class restricted to the singleton of the argument is a function. (Contributed by Alexander van der Vekens, 26-May-2017.) |
Ref | Expression |
---|---|
aovvfunressn | ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 → Fun (𝐹 ↾ {〈𝐴, 𝐵〉})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-aov 39847 | . . 3 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''〈𝐴, 𝐵〉) | |
2 | 1 | eleq1i 2679 | . 2 ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 ↔ (𝐹'''〈𝐴, 𝐵〉) ∈ 𝐶) |
3 | afvvfunressn 39872 | . 2 ⊢ ((𝐹'''〈𝐴, 𝐵〉) ∈ 𝐶 → Fun (𝐹 ↾ {〈𝐴, 𝐵〉})) | |
4 | 2, 3 | sylbi 206 | 1 ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 → Fun (𝐹 ↾ {〈𝐴, 𝐵〉})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 {csn 4125 〈cop 4131 ↾ cres 5040 Fun wfun 5798 '''cafv 39843 ((caov 39844 |
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-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 |
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 df-aov 39847 |
This theorem is referenced by: (None) |
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