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Mirrors > Home > MPE Home > Th. List > mapval | Structured version Visualization version GIF version |
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 8846 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ↑m 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴}) | |
4 | 1, 2, 3 | mp2an 691 | 1 ⊢ (𝐴 ↑m 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 {cab 2704 Vcvv 3469 ⟶wf 6538 (class class class)co 7414 ↑m cmap 8836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-map 8838 |
This theorem is referenced by: 0map0sn0 8895 maprnin 32497 poimirlem4 37032 poimirlem9 37037 poimirlem26 37054 poimirlem27 37055 poimirlem28 37056 poimirlem32 37060 lautset 39492 pautsetN 39508 tendoset 40169 |
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