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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 44260 | . . 3 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹'''𝐴) = V) | |
2 | nvelim 44241 | . . 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 1543 ∈ wcel 2110 Vcvv 3401 {csn 4531 ↾ cres 5542 Fun wfun 6363 '''cafv 44235 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-int 4850 df-br 5044 df-opab 5106 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-res 5552 df-iota 6327 df-fun 6371 df-fv 6377 df-aiota 44203 df-dfat 44237 df-afv 44238 |
This theorem is referenced by: aovvfunressn 44305 |
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