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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 41198 | . . 3 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''〈𝐴, 𝐵〉) | |
2 | 1 | eleq1i 2692 | . 2 ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 ↔ (𝐹'''〈𝐴, 𝐵〉) ∈ 𝐶) |
3 | afvvfunressn 41223 | . 2 ⊢ ((𝐹'''〈𝐴, 𝐵〉) ∈ 𝐶 → Fun (𝐹 ↾ {〈𝐴, 𝐵〉})) | |
4 | 2, 3 | sylbi 207 | 1 ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 → Fun (𝐹 ↾ {〈𝐴, 𝐵〉})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 {csn 4177 〈cop 4183 ↾ cres 5116 Fun wfun 5882 '''cafv 41194 ((caov 41195 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-un 3579 df-if 4087 df-fv 5896 df-dfat 41196 df-afv 41197 df-aov 41198 |
This theorem is referenced by: (None) |
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