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Mirrors > Home > MPE Home > Th. List > Mathboxes > funeldmb | Structured version Visualization version GIF version |
Description: If ∅ is not part of the range of a function 𝐹, then 𝐴 is in the domain of 𝐹 iff (𝐹‘𝐴) ≠ ∅. (Contributed by Scott Fenton, 7-Dec-2021.) |
Ref | Expression |
---|---|
funeldmb | ⊢ ((Fun 𝐹 ∧ ¬ ∅ ∈ ran 𝐹) → (𝐴 ∈ dom 𝐹 ↔ (𝐹‘𝐴) ≠ ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvelrn 6602 | . . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹) | |
2 | 1 | ex 403 | . . . . . . 7 ⊢ (Fun 𝐹 → (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) ∈ ran 𝐹)) |
3 | 2 | adantr 474 | . . . . . 6 ⊢ ((Fun 𝐹 ∧ (𝐹‘𝐴) = ∅) → (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) ∈ ran 𝐹)) |
4 | eleq1 2895 | . . . . . . 7 ⊢ ((𝐹‘𝐴) = ∅ → ((𝐹‘𝐴) ∈ ran 𝐹 ↔ ∅ ∈ ran 𝐹)) | |
5 | 4 | adantl 475 | . . . . . 6 ⊢ ((Fun 𝐹 ∧ (𝐹‘𝐴) = ∅) → ((𝐹‘𝐴) ∈ ran 𝐹 ↔ ∅ ∈ ran 𝐹)) |
6 | 3, 5 | sylibd 231 | . . . . 5 ⊢ ((Fun 𝐹 ∧ (𝐹‘𝐴) = ∅) → (𝐴 ∈ dom 𝐹 → ∅ ∈ ran 𝐹)) |
7 | 6 | con3d 150 | . . . 4 ⊢ ((Fun 𝐹 ∧ (𝐹‘𝐴) = ∅) → (¬ ∅ ∈ ran 𝐹 → ¬ 𝐴 ∈ dom 𝐹)) |
8 | 7 | impancom 445 | . . 3 ⊢ ((Fun 𝐹 ∧ ¬ ∅ ∈ ran 𝐹) → ((𝐹‘𝐴) = ∅ → ¬ 𝐴 ∈ dom 𝐹)) |
9 | ndmfv 6464 | . . 3 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅) | |
10 | 8, 9 | impbid1 217 | . 2 ⊢ ((Fun 𝐹 ∧ ¬ ∅ ∈ ran 𝐹) → ((𝐹‘𝐴) = ∅ ↔ ¬ 𝐴 ∈ dom 𝐹)) |
11 | 10 | necon2abid 3042 | 1 ⊢ ((Fun 𝐹 ∧ ¬ ∅ ∈ ran 𝐹) → (𝐴 ∈ dom 𝐹 ↔ (𝐹‘𝐴) ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ≠ wne 3000 ∅c0 4145 dom cdm 5343 ran crn 5344 Fun wfun 6118 ‘cfv 6124 |
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-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 |
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 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-sbc 3664 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-id 5251 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-iota 6087 df-fun 6126 df-fn 6127 df-fv 6132 |
This theorem is referenced by: nosepssdm 32376 |
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