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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 6920 | . . . . 5 ⊢ (𝐹‘𝐴) ∈ V | |
2 | 1 | unisn 4931 | . . . 4 ⊢ ∪ {(𝐹‘𝐴)} = (𝐹‘𝐴) |
3 | eqid 2735 | . . . . . . 7 ⊢ dom 𝐹 = dom 𝐹 | |
4 | df-fn 6566 | . . . . . . 7 ⊢ (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 ∧ dom 𝐹 = dom 𝐹)) | |
5 | 3, 4 | mpbiran2 710 | . . . . . 6 ⊢ (𝐹 Fn dom 𝐹 ↔ Fun 𝐹) |
6 | fnsnfv 6988 | . . . . . 6 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐴 ∈ dom 𝐹) → {(𝐹‘𝐴)} = (𝐹 “ {𝐴})) | |
7 | 5, 6 | sylanbr 582 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → {(𝐹‘𝐴)} = (𝐹 “ {𝐴})) |
8 | 7 | unieqd 4925 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ∪ {(𝐹‘𝐴)} = ∪ (𝐹 “ {𝐴})) |
9 | 2, 8 | eqtr3id 2789 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴})) |
10 | 9 | ex 412 | . 2 ⊢ (Fun 𝐹 → (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴}))) |
11 | ndmfv 6942 | . . 3 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅) | |
12 | ndmima 6124 | . . . . 5 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹 “ {𝐴}) = ∅) | |
13 | 12 | unieqd 4925 | . . . 4 ⊢ (¬ 𝐴 ∈ dom 𝐹 → ∪ (𝐹 “ {𝐴}) = ∪ ∅) |
14 | uni0 4940 | . . . 4 ⊢ ∪ ∅ = ∅ | |
15 | 13, 14 | eqtrdi 2791 | . . 3 ⊢ (¬ 𝐴 ∈ dom 𝐹 → ∪ (𝐹 “ {𝐴}) = ∅) |
16 | 11, 15 | eqtr4d 2778 | . 2 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴})) |
17 | 10, 16 | pm2.61d1 180 | 1 ⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∅c0 4339 {csn 4631 ∪ cuni 4912 dom cdm 5689 “ cima 5692 Fun wfun 6557 Fn wfn 6558 ‘cfv 6563 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-fv 6571 |
This theorem is referenced by: funfv2 6997 fvun 6999 dffv2 7004 setrecsss 48932 |
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