mapval - Metamath Proof Explorer

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Theorem mapval 8863

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.)

**Hypotheses**
Ref	Expression
mapval.1	⊢ 𝐴 ∈ V
mapval.2	⊢ 𝐵 ∈ V

**Assertion**
Ref	Expression
mapval	⊢ (𝐴 ↑_m 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴}

Distinct variable groups: 𝐴,𝑓 𝐵,𝑓

**Proof of Theorem mapval**
Step	Hyp	Ref	Expression
1		mapval.1	. 2 ⊢ 𝐴 ∈ V
2		mapval.2	. 2 ⊢ 𝐵 ∈ V
3		mapvalg 8861	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ↑_m 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴})
4	1, 2, 3	mp2an 690	1 ⊢ (𝐴 ↑_m 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴}

Colors of variables: wff setvar class

Syntax hints: = wceq 1533 ∈ wcel 2098 {cab 2705 Vcvv 3473 ⟶wf 6549 (class class class)co 7426 ↑_m cmap 8851

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-map 8853

This theorem is referenced by: 0map0sn0 8910 maprnin 32534 poimirlem4 37130 poimirlem9 37135 poimirlem26 37152 poimirlem27 37153 poimirlem28 37154 poimirlem32 37158 lautset 39587 pautsetN 39603 tendoset 40264

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