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Mirrors > Home > MPE Home > Th. List > Mathboxes > afvvfunressn | Structured version Visualization version GIF version |
Description: If the function 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, 25-May-2017.) |
Ref | Expression |
---|---|
afvvfunressn | ⊢ ((𝐹'''𝐴) ∈ 𝐵 → Fun (𝐹 ↾ {𝐴})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfunsnafv 43361 | . . 3 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹'''𝐴) = V) | |
2 | nvelim 43342 | . . 3 ⊢ ((𝐹'''𝐴) = V → ¬ (𝐹'''𝐴) ∈ 𝐵) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → ¬ (𝐹'''𝐴) ∈ 𝐵) |
4 | 3 | con4i 114 | 1 ⊢ ((𝐹'''𝐴) ∈ 𝐵 → Fun (𝐹 ↾ {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3494 {csn 4567 ↾ cres 5557 Fun wfun 6349 '''cafv 43336 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-int 4877 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-res 5567 df-iota 6314 df-fun 6357 df-fv 6363 df-aiota 43305 df-dfat 43338 df-afv 43339 |
This theorem is referenced by: aovvfunressn 43406 |
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