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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 8527 (which does not require ax-rep 5188) 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 8545), 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 3419 | . . . 4 ⊢ 𝑚 ∈ V | |
2 | feq1 6535 | . . . 4 ⊢ (𝑓 = 𝑚 → (𝑓:𝐴⟶𝐵 ↔ 𝑚:𝐴⟶𝐵)) | |
3 | 1, 2 | elab 3594 | . . 3 ⊢ (𝑚 ∈ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ↔ 𝑚:𝐴⟶𝐵) |
4 | simpr 488 | . . . . . . 7 ⊢ ((𝑚:𝐴⟶𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉) | |
5 | dmfex 7694 | . . . . . . . . 9 ⊢ ((𝑚 ∈ V ∧ 𝑚:𝐴⟶𝐵) → 𝐴 ∈ V) | |
6 | 1, 5 | mpan 690 | . . . . . . . 8 ⊢ (𝑚:𝐴⟶𝐵 → 𝐴 ∈ V) |
7 | 6 | adantr 484 | . . . . . . 7 ⊢ ((𝑚:𝐴⟶𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ V) |
8 | 4, 7 | elmapd 8531 | . . . . . 6 ⊢ ((𝑚:𝐴⟶𝐵 ∧ 𝐵 ∈ 𝑉) → (𝑚 ∈ (𝐵 ↑m 𝐴) ↔ 𝑚:𝐴⟶𝐵)) |
9 | 8 | exbiri 811 | . . . . 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 8539 | . . . 4 ⊢ (𝑚 ∈ (𝐵 ↑m 𝐴) → 𝑚:𝐴⟶𝐵) | |
12 | 10, 11 | impbid1 228 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (𝑚:𝐴⟶𝐵 ↔ 𝑚 ∈ (𝐵 ↑m 𝐴))) |
13 | 3, 12 | syl5bb 286 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝑚 ∈ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ↔ 𝑚 ∈ (𝐵 ↑m 𝐴))) |
14 | 13 | eqrdv 2736 | 1 ⊢ (𝐵 ∈ 𝑉 → {𝑓 ∣ 𝑓:𝐴⟶𝐵} = (𝐵 ↑m 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2111 {cab 2715 Vcvv 3415 ⟶wf 6385 (class class class)co 7222 ↑m cmap 8517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5201 ax-nul 5208 ax-pow 5267 ax-pr 5331 ax-un 7532 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3417 df-sbc 3704 df-csb 3821 df-dif 3878 df-un 3880 df-in 3882 df-ss 3892 df-nul 4247 df-if 4449 df-pw 4524 df-sn 4551 df-pr 4553 df-op 4557 df-uni 4829 df-iun 4915 df-br 5063 df-opab 5125 df-mpt 5145 df-id 5464 df-xp 5566 df-rel 5567 df-cnv 5568 df-co 5569 df-dm 5570 df-rn 5571 df-res 5572 df-ima 5573 df-iota 6347 df-fun 6391 df-fn 6392 df-f 6393 df-fv 6397 df-ov 7225 df-oprab 7226 df-mpo 7227 df-1st 7770 df-2nd 7771 df-map 8519 |
This theorem is referenced by: mapssfset 8541 fsetex 8546 |
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