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Mirrors > Home > MPE Home > Th. List > mapfset | Structured version Visualization version GIF version |
Description: If 𝐵 is a set, the value of the set exponentiation (𝐵 ↑m 𝐴) is the class of all functions from 𝐴 to 𝐵. Generalisation of mapvalg 8829 (which does not require ax-rep 5278) to arbitrary domains. Note that the class {𝑓 ∣ 𝑓:𝐴⟶𝐵} can only contain set-functions, as opposed to arbitrary class-functions. When 𝐴 is a proper class, there can be no set-functions on it, so the above class is empty (see also fsetdmprc0 8848), hence a set. In this case, both sides of the equality in this theorem are the empty set. (Contributed by AV, 8-Aug-2024.) |
Ref | Expression |
---|---|
mapfset | ⊢ (𝐵 ∈ 𝑉 → {𝑓 ∣ 𝑓:𝐴⟶𝐵} = (𝐵 ↑m 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3472 | . . . 4 ⊢ 𝑚 ∈ V | |
2 | feq1 6691 | . . . 4 ⊢ (𝑓 = 𝑚 → (𝑓:𝐴⟶𝐵 ↔ 𝑚:𝐴⟶𝐵)) | |
3 | 1, 2 | elab 3663 | . . 3 ⊢ (𝑚 ∈ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ↔ 𝑚:𝐴⟶𝐵) |
4 | simpr 484 | . . . . . . 7 ⊢ ((𝑚:𝐴⟶𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉) | |
5 | dmfex 7894 | . . . . . . . . 9 ⊢ ((𝑚 ∈ V ∧ 𝑚:𝐴⟶𝐵) → 𝐴 ∈ V) | |
6 | 1, 5 | mpan 687 | . . . . . . . 8 ⊢ (𝑚:𝐴⟶𝐵 → 𝐴 ∈ V) |
7 | 6 | adantr 480 | . . . . . . 7 ⊢ ((𝑚:𝐴⟶𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ V) |
8 | 4, 7 | elmapd 8833 | . . . . . 6 ⊢ ((𝑚:𝐴⟶𝐵 ∧ 𝐵 ∈ 𝑉) → (𝑚 ∈ (𝐵 ↑m 𝐴) ↔ 𝑚:𝐴⟶𝐵)) |
9 | 8 | exbiri 808 | . . . . 5 ⊢ (𝑚:𝐴⟶𝐵 → (𝐵 ∈ 𝑉 → (𝑚:𝐴⟶𝐵 → 𝑚 ∈ (𝐵 ↑m 𝐴)))) |
10 | 9 | pm2.43b 55 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝑚:𝐴⟶𝐵 → 𝑚 ∈ (𝐵 ↑m 𝐴))) |
11 | elmapi 8842 | . . . 4 ⊢ (𝑚 ∈ (𝐵 ↑m 𝐴) → 𝑚:𝐴⟶𝐵) | |
12 | 10, 11 | impbid1 224 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (𝑚:𝐴⟶𝐵 ↔ 𝑚 ∈ (𝐵 ↑m 𝐴))) |
13 | 3, 12 | bitrid 283 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝑚 ∈ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ↔ 𝑚 ∈ (𝐵 ↑m 𝐴))) |
14 | 13 | eqrdv 2724 | 1 ⊢ (𝐵 ∈ 𝑉 → {𝑓 ∣ 𝑓:𝐴⟶𝐵} = (𝐵 ↑m 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 {cab 2703 Vcvv 3468 ⟶wf 6532 (class class class)co 7404 ↑m cmap 8819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-map 8821 |
This theorem is referenced by: mapssfset 8844 fsetex 8849 |
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