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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's a function. (Contributed by NM, 22-May-1998.) |
Ref | Expression |
---|---|
funfv | ⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6113 | . . . . 5 ⊢ (𝐹‘𝐴) ∈ V | |
2 | 1 | unisn 4387 | . . . 4 ⊢ ∪ {(𝐹‘𝐴)} = (𝐹‘𝐴) |
3 | eqid 2610 | . . . . . . 7 ⊢ dom 𝐹 = dom 𝐹 | |
4 | df-fn 5807 | . . . . . . 7 ⊢ (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 ∧ dom 𝐹 = dom 𝐹)) | |
5 | 3, 4 | mpbiran2 956 | . . . . . 6 ⊢ (𝐹 Fn dom 𝐹 ↔ Fun 𝐹) |
6 | fnsnfv 6168 | . . . . . 6 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐴 ∈ dom 𝐹) → {(𝐹‘𝐴)} = (𝐹 “ {𝐴})) | |
7 | 5, 6 | sylanbr 489 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → {(𝐹‘𝐴)} = (𝐹 “ {𝐴})) |
8 | 7 | unieqd 4382 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ∪ {(𝐹‘𝐴)} = ∪ (𝐹 “ {𝐴})) |
9 | 2, 8 | syl5eqr 2658 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴})) |
10 | 9 | ex 449 | . 2 ⊢ (Fun 𝐹 → (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴}))) |
11 | ndmfv 6128 | . . 3 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅) | |
12 | ndmima 5421 | . . . . 5 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹 “ {𝐴}) = ∅) | |
13 | 12 | unieqd 4382 | . . . 4 ⊢ (¬ 𝐴 ∈ dom 𝐹 → ∪ (𝐹 “ {𝐴}) = ∪ ∅) |
14 | uni0 4401 | . . . 4 ⊢ ∪ ∅ = ∅ | |
15 | 13, 14 | syl6eq 2660 | . . 3 ⊢ (¬ 𝐴 ∈ dom 𝐹 → ∪ (𝐹 “ {𝐴}) = ∅) |
16 | 11, 15 | eqtr4d 2647 | . 2 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴})) |
17 | 10, 16 | pm2.61d1 170 | 1 ⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∅c0 3874 {csn 4125 ∪ cuni 4372 dom cdm 5038 “ cima 5041 Fun wfun 5798 Fn wfn 5799 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 |
This theorem is referenced by: funfv2 6176 fvun 6178 dffv2 6181 |
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