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| Description: The value of set exponentiation (inference version). (𝐴 ↑m 𝐵) is the set of all functions that map from 𝐵 to 𝐴. Definition 10.24 of [Kunen] p. 24. (Contributed by NM, 8-Dec-2003.) | 
| Ref | Expression | 
|---|---|
| mapval.1 | ⊢ 𝐴 ∈ V | 
| mapval.2 | ⊢ 𝐵 ∈ V | 
| Ref | Expression | 
|---|---|
| mapval | ⊢ (𝐴 ↑m 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴} | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | mapval.1 | . 2 ⊢ 𝐴 ∈ V | |
| 2 | mapval.2 | . 2 ⊢ 𝐵 ∈ V | |
| 3 | mapvalg 8876 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ↑m 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴}) | |
| 4 | 1, 2, 3 | mp2an 692 | 1 ⊢ (𝐴 ↑m 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴} | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∈ wcel 2108 {cab 2714 Vcvv 3480 ⟶wf 6557 (class class class)co 7431 ↑m cmap 8866 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 | 
| This theorem is referenced by: 0map0sn0 8925 maprnin 32742 poimirlem4 37631 poimirlem9 37636 poimirlem26 37653 poimirlem27 37654 poimirlem28 37655 poimirlem32 37659 lautset 40084 pautsetN 40100 tendoset 40761 | 
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