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Theorem mapfset 8843
Description: If 𝐵 is a set, the value of the set exponentiation (𝐵m 𝐴) is the class of all functions from 𝐴 to 𝐵. Generalisation of mapvalg 8829 (which does not require ax-rep 5278) to arbitrary domains. Note that the class {𝑓𝑓:𝐴𝐵} can only contain set-functions, as opposed to arbitrary class-functions. When 𝐴 is a proper class, there can be no set-functions on it, so the above class is empty (see also fsetdmprc0 8848), hence a set. In this case, both sides of the equality in this theorem are the empty set. (Contributed by AV, 8-Aug-2024.)
Assertion
Ref Expression
mapfset (𝐵𝑉 → {𝑓𝑓:𝐴𝐵} = (𝐵m 𝐴))
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓
Allowed substitution hint:   𝑉(𝑓)

Proof of Theorem mapfset
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 vex 3472 . . . 4 𝑚 ∈ V
2 feq1 6691 . . . 4 (𝑓 = 𝑚 → (𝑓:𝐴𝐵𝑚:𝐴𝐵))
31, 2elab 3663 . . 3 (𝑚 ∈ {𝑓𝑓:𝐴𝐵} ↔ 𝑚:𝐴𝐵)
4 simpr 484 . . . . . . 7 ((𝑚:𝐴𝐵𝐵𝑉) → 𝐵𝑉)
5 dmfex 7894 . . . . . . . . 9 ((𝑚 ∈ V ∧ 𝑚:𝐴𝐵) → 𝐴 ∈ V)
61, 5mpan 687 . . . . . . . 8 (𝑚:𝐴𝐵𝐴 ∈ V)
76adantr 480 . . . . . . 7 ((𝑚:𝐴𝐵𝐵𝑉) → 𝐴 ∈ V)
84, 7elmapd 8833 . . . . . 6 ((𝑚:𝐴𝐵𝐵𝑉) → (𝑚 ∈ (𝐵m 𝐴) ↔ 𝑚:𝐴𝐵))
98exbiri 808 . . . . 5 (𝑚:𝐴𝐵 → (𝐵𝑉 → (𝑚:𝐴𝐵𝑚 ∈ (𝐵m 𝐴))))
109pm2.43b 55 . . . 4 (𝐵𝑉 → (𝑚:𝐴𝐵𝑚 ∈ (𝐵m 𝐴)))
11 elmapi 8842 . . . 4 (𝑚 ∈ (𝐵m 𝐴) → 𝑚:𝐴𝐵)
1210, 11impbid1 224 . . 3 (𝐵𝑉 → (𝑚:𝐴𝐵𝑚 ∈ (𝐵m 𝐴)))
133, 12bitrid 283 . 2 (𝐵𝑉 → (𝑚 ∈ {𝑓𝑓:𝐴𝐵} ↔ 𝑚 ∈ (𝐵m 𝐴)))
1413eqrdv 2724 1 (𝐵𝑉 → {𝑓𝑓:𝐴𝐵} = (𝐵m 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  {cab 2703  Vcvv 3468  wf 6532  (class class class)co 7404  m cmap 8819
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-map 8821
This theorem is referenced by:  mapssfset  8844  fsetex  8849
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