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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfatelrn | Structured version Visualization version GIF version |
Description: The value of a function 𝐹 at a set 𝐴 is in the range of the function 𝐹 if 𝐹 is defined at 𝐴. (Contributed by AV, 1-Sep-2022.) |
Ref | Expression |
---|---|
dfatelrn | ⊢ (𝐹 defAt 𝐴 → (𝐹‘𝐴) ∈ ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dfat 42020 | . 2 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | |
2 | funressndmfvrn 41974 | . . 3 ⊢ ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹) | |
3 | 2 | ancoms 452 | . 2 ⊢ ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐹‘𝐴) ∈ ran 𝐹) |
4 | 1, 3 | sylbi 209 | 1 ⊢ (𝐹 defAt 𝐴 → (𝐹‘𝐴) ∈ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2166 {csn 4396 dom cdm 5341 ran crn 5342 ↾ cres 5343 Fun wfun 6116 ‘cfv 6122 defAt wdfat 42017 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-sep 5004 ax-nul 5012 ax-pr 5126 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ral 3121 df-rex 3122 df-rab 3125 df-v 3415 df-sbc 3662 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-nul 4144 df-if 4306 df-sn 4397 df-pr 4399 df-op 4403 df-uni 4658 df-br 4873 df-opab 4935 df-id 5249 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-iota 6085 df-fun 6124 df-fn 6125 df-fv 6130 df-dfat 42020 |
This theorem is referenced by: (None) |
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