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| Mirrors > Home > MPE Home > Th. List > fvfundmfvn0 | Structured version Visualization version GIF version | ||
| Description: If the "value of a class" at an argument is not the empty set, then the argument is in the domain of the class and the class restricted to the singleton formed on that argument is a function. (Contributed by Alexander van der Vekens, 26-May-2017.) (Proof shortened by BJ, 13-Aug-2022.) | 
| Ref | Expression | 
|---|---|
| fvfundmfvn0 | ⊢ ((𝐹‘𝐴) ≠ ∅ → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ndmfv 6941 | . . 3 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅) | |
| 2 | 1 | necon1ai 2968 | . 2 ⊢ ((𝐹‘𝐴) ≠ ∅ → 𝐴 ∈ dom 𝐹) | 
| 3 | nfunsn 6948 | . . 3 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹‘𝐴) = ∅) | |
| 4 | 3 | necon1ai 2968 | . 2 ⊢ ((𝐹‘𝐴) ≠ ∅ → Fun (𝐹 ↾ {𝐴})) | 
| 5 | 2, 4 | jca 511 | 1 ⊢ ((𝐹‘𝐴) ≠ ∅ → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ≠ wne 2940 ∅c0 4333 {csn 4626 dom cdm 5685 ↾ cres 5687 Fun wfun 6555 ‘cfv 6561 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-res 5697 df-iota 6514 df-fun 6563 df-fv 6569 | 
| This theorem is referenced by: fvn0ssdmfun 7094 feldmfvelcdm 7106 fvn0fvelrnOLD 7183 umgrnloopv 29123 usgrnloopvALT 29218 afvpcfv0 47158 afvfvn0fveq 47162 afv0nbfvbi 47163 afv2fvn0fveq 47276 ovn0dmfun 48072 | 
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