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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > afvelrn | Structured version Visualization version GIF version |
Description: A function's value belongs to its range, analogous to fvelrn 7079. (Contributed by Alexander van der Vekens, 25-May-2017.) |
Ref | Expression |
---|---|
afvelrn | ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹'''𝐴) ∈ ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funres 6591 | . . . . . 6 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ {𝐴})) | |
2 | 1 | anim1i 616 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹)) |
3 | 2 | ancomd 463 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) |
4 | df-dfat 45827 | . . . 4 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | |
5 | 3, 4 | sylibr 233 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 𝐹 defAt 𝐴) |
6 | afvfundmfveq 45846 | . . . 4 ⊢ (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹‘𝐴)) | |
7 | 6 | eqcomd 2739 | . . 3 ⊢ (𝐹 defAt 𝐴 → (𝐹‘𝐴) = (𝐹'''𝐴)) |
8 | 5, 7 | syl 17 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) = (𝐹'''𝐴)) |
9 | fvelrn 7079 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹) | |
10 | 8, 9 | eqeltrrd 2835 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹'''𝐴) ∈ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {csn 4629 dom cdm 5677 ran crn 5678 ↾ cres 5679 Fun wfun 6538 ‘cfv 6544 defAt wdfat 45824 '''cafv 45825 |
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 5300 ax-nul 5307 ax-pr 5428 |
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-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-iota 6496 df-fun 6546 df-fn 6547 df-fv 6552 df-aiota 45793 df-dfat 45827 df-afv 45828 |
This theorem is referenced by: fnafvelrn 45877 |
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