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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 6685 | . . . . 5 ⊢ (𝐹‘𝐴) ∈ V | |
2 | 1 | unisn 4860 | . . . 4 ⊢ ∪ {(𝐹‘𝐴)} = (𝐹‘𝐴) |
3 | eqid 2823 | . . . . . . 7 ⊢ dom 𝐹 = dom 𝐹 | |
4 | df-fn 6360 | . . . . . . 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 6745 | . . . . . 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 4854 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ∪ {(𝐹‘𝐴)} = ∪ (𝐹 “ {𝐴})) |
9 | 2, 8 | syl5eqr 2872 | . . 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 6702 | . . 3 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅) | |
12 | ndmima 5968 | . . . . 5 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹 “ {𝐴}) = ∅) | |
13 | 12 | unieqd 4854 | . . . 4 ⊢ (¬ 𝐴 ∈ dom 𝐹 → ∪ (𝐹 “ {𝐴}) = ∪ ∅) |
14 | uni0 4868 | . . . 4 ⊢ ∪ ∅ = ∅ | |
15 | 13, 14 | syl6eq 2874 | . . 3 ⊢ (¬ 𝐴 ∈ dom 𝐹 → ∪ (𝐹 “ {𝐴}) = ∅) |
16 | 11, 15 | eqtr4d 2861 | . 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 1537 ∈ wcel 2114 ∅c0 4293 {csn 4569 ∪ cuni 4840 dom cdm 5557 “ cima 5560 Fun wfun 6351 Fn wfn 6352 ‘cfv 6357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-fv 6365 |
This theorem is referenced by: funfv2 6753 fvun 6755 dffv2 6758 setrecsss 44810 |
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