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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 6935 | . 2 ⊢ (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅}) | |
| 2 | nelsn 4665 | . 2 ⊢ ((𝐹‘𝑋) ≠ ∅ → ¬ (𝐹‘𝑋) ∈ {∅}) | |
| 3 | elunnel2 4154 | . 2 ⊢ (((𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅}) ∧ ¬ (𝐹‘𝑋) ∈ {∅}) → (𝐹‘𝑋) ∈ ran 𝐹) | |
| 4 | 1, 2, 3 | sylancr 587 | 1 ⊢ ((𝐹‘𝑋) ≠ ∅ → (𝐹‘𝑋) ∈ ran 𝐹) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2107 ≠ wne 2939 ∪ cun 3948 ∅c0 4332 {csn 4625 ran crn 5685 ‘cfv 6560 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-cnv 5692 df-dm 5694 df-rn 5695 df-iota 6513 df-fv 6568 | 
| This theorem is referenced by: elfvunirn 6937 wlkvtxiedg 29644 | 
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