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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 6873 | . 2 ⊢ (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅}) | |
2 | nelsn 4627 | . 2 ⊢ ((𝐹‘𝑋) ≠ ∅ → ¬ (𝐹‘𝑋) ∈ {∅}) | |
3 | elunnel2 4111 | . 2 ⊢ (((𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅}) ∧ ¬ (𝐹‘𝑋) ∈ {∅}) → (𝐹‘𝑋) ∈ ran 𝐹) | |
4 | 1, 2, 3 | sylancr 588 | 1 ⊢ ((𝐹‘𝑋) ≠ ∅ → (𝐹‘𝑋) ∈ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2107 ≠ wne 2940 ∪ cun 3909 ∅c0 4283 {csn 4587 ran crn 5635 ‘cfv 6497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-cnv 5642 df-dm 5644 df-rn 5645 df-iota 6449 df-fv 6505 |
This theorem is referenced by: elfvunirn 6875 wlkvtxiedg 28615 |
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