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Mirrors > Home > MPE Home > Th. List > fvn0fvelrn | Structured version Visualization version GIF version |
Description: If the value of a function is not null, the value is an element of the range of the function. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Proof shortened by SN, 13-Jan-2025.) |
Ref | Expression |
---|---|
fvn0fvelrn | ⊢ ((𝐹‘𝑋) ≠ ∅ → (𝐹‘𝑋) ∈ ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvrn0 6922 | . 2 ⊢ (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅}) | |
2 | nelsn 4664 | . 2 ⊢ ((𝐹‘𝑋) ≠ ∅ → ¬ (𝐹‘𝑋) ∈ {∅}) | |
3 | elunnel2 4143 | . 2 ⊢ (((𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅}) ∧ ¬ (𝐹‘𝑋) ∈ {∅}) → (𝐹‘𝑋) ∈ ran 𝐹) | |
4 | 1, 2, 3 | sylancr 585 | 1 ⊢ ((𝐹‘𝑋) ≠ ∅ → (𝐹‘𝑋) ∈ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2098 ≠ wne 2930 ∪ cun 3937 ∅c0 4318 {csn 4624 ran crn 5673 ‘cfv 6543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-cnv 5680 df-dm 5682 df-rn 5683 df-iota 6495 df-fv 6551 |
This theorem is referenced by: elfvunirn 6924 wlkvtxiedg 29483 |
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