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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 8405 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ↑m 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴}) | |
4 | 1, 2, 3 | mp2an 688 | 1 ⊢ (𝐴 ↑m 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 {cab 2796 Vcvv 3492 ⟶wf 6344 (class class class)co 7145 ↑m cmap 8395 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-map 8397 |
This theorem is referenced by: maprnin 30393 poimirlem4 34777 poimirlem9 34782 poimirlem26 34799 poimirlem27 34800 poimirlem28 34801 poimirlem32 34805 lautset 37098 pautsetN 37114 tendoset 37775 efmndbas0 43988 |
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