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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 8447 | . 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 1542 ∈ wcel 2114 {cab 2716 Vcvv 3398 ⟶wf 6335 (class class class)co 7170 ↑m cmap 8437 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-fv 6347 df-ov 7173 df-oprab 7174 df-mpo 7175 df-map 8439 |
This theorem is referenced by: 0map0sn0 8495 maprnin 30641 poimirlem4 35404 poimirlem9 35409 poimirlem26 35426 poimirlem27 35427 poimirlem28 35428 poimirlem32 35432 lautset 37719 pautsetN 37735 tendoset 38396 |
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