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Mirrors > Home > MPE Home > Th. List > funfv | Structured version Visualization version GIF version |
Description: A simplified expression for the value of a function when we know it is a function. (Contributed by NM, 22-May-1998.) |
Ref | Expression |
---|---|
funfv | ⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6682 | . . . . 5 ⊢ (𝐹‘𝐴) ∈ V | |
2 | 1 | unisn 4857 | . . . 4 ⊢ ∪ {(𝐹‘𝐴)} = (𝐹‘𝐴) |
3 | eqid 2821 | . . . . . . 7 ⊢ dom 𝐹 = dom 𝐹 | |
4 | df-fn 6357 | . . . . . . 7 ⊢ (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 ∧ dom 𝐹 = dom 𝐹)) | |
5 | 3, 4 | mpbiran2 708 | . . . . . 6 ⊢ (𝐹 Fn dom 𝐹 ↔ Fun 𝐹) |
6 | fnsnfv 6742 | . . . . . 6 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐴 ∈ dom 𝐹) → {(𝐹‘𝐴)} = (𝐹 “ {𝐴})) | |
7 | 5, 6 | sylanbr 584 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → {(𝐹‘𝐴)} = (𝐹 “ {𝐴})) |
8 | 7 | unieqd 4851 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ∪ {(𝐹‘𝐴)} = ∪ (𝐹 “ {𝐴})) |
9 | 2, 8 | syl5eqr 2870 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴})) |
10 | 9 | ex 415 | . 2 ⊢ (Fun 𝐹 → (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴}))) |
11 | ndmfv 6699 | . . 3 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅) | |
12 | ndmima 5965 | . . . . 5 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹 “ {𝐴}) = ∅) | |
13 | 12 | unieqd 4851 | . . . 4 ⊢ (¬ 𝐴 ∈ dom 𝐹 → ∪ (𝐹 “ {𝐴}) = ∪ ∅) |
14 | uni0 4865 | . . . 4 ⊢ ∪ ∅ = ∅ | |
15 | 13, 14 | syl6eq 2872 | . . 3 ⊢ (¬ 𝐴 ∈ dom 𝐹 → ∪ (𝐹 “ {𝐴}) = ∅) |
16 | 11, 15 | eqtr4d 2859 | . 2 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴})) |
17 | 10, 16 | pm2.61d1 182 | 1 ⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∅c0 4290 {csn 4566 ∪ cuni 4837 dom cdm 5554 “ cima 5557 Fun wfun 6348 Fn wfn 6349 ‘cfv 6354 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-fv 6362 |
This theorem is referenced by: funfv2 6750 fvun 6752 dffv2 6755 setrecsss 44802 |
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