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Theorem mapfset 8540
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 8527 (which does not require ax-rep 5188) 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 8545), 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 3419 . . . 4 𝑚 ∈ V
2 feq1 6535 . . . 4 (𝑓 = 𝑚 → (𝑓:𝐴𝐵𝑚:𝐴𝐵))
31, 2elab 3594 . . 3 (𝑚 ∈ {𝑓𝑓:𝐴𝐵} ↔ 𝑚:𝐴𝐵)
4 simpr 488 . . . . . . 7 ((𝑚:𝐴𝐵𝐵𝑉) → 𝐵𝑉)
5 dmfex 7694 . . . . . . . . 9 ((𝑚 ∈ V ∧ 𝑚:𝐴𝐵) → 𝐴 ∈ V)
61, 5mpan 690 . . . . . . . 8 (𝑚:𝐴𝐵𝐴 ∈ V)
76adantr 484 . . . . . . 7 ((𝑚:𝐴𝐵𝐵𝑉) → 𝐴 ∈ V)
84, 7elmapd 8531 . . . . . 6 ((𝑚:𝐴𝐵𝐵𝑉) → (𝑚 ∈ (𝐵m 𝐴) ↔ 𝑚:𝐴𝐵))
98exbiri 811 . . . . 5 (𝑚:𝐴𝐵 → (𝐵𝑉 → (𝑚:𝐴𝐵𝑚 ∈ (𝐵m 𝐴))))
109pm2.43b 55 . . . 4 (𝐵𝑉 → (𝑚:𝐴𝐵𝑚 ∈ (𝐵m 𝐴)))
11 elmapi 8539 . . . 4 (𝑚 ∈ (𝐵m 𝐴) → 𝑚:𝐴𝐵)
1210, 11impbid1 228 . . 3 (𝐵𝑉 → (𝑚:𝐴𝐵𝑚 ∈ (𝐵m 𝐴)))
133, 12syl5bb 286 . 2 (𝐵𝑉 → (𝑚 ∈ {𝑓𝑓:𝐴𝐵} ↔ 𝑚 ∈ (𝐵m 𝐴)))
1413eqrdv 2736 1 (𝐵𝑉 → {𝑓𝑓:𝐴𝐵} = (𝐵m 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2111  {cab 2715  Vcvv 3415  wf 6385  (class class class)co 7222  m cmap 8517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5201  ax-nul 5208  ax-pow 5267  ax-pr 5331  ax-un 7532
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3417  df-sbc 3704  df-csb 3821  df-dif 3878  df-un 3880  df-in 3882  df-ss 3892  df-nul 4247  df-if 4449  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4829  df-iun 4915  df-br 5063  df-opab 5125  df-mpt 5145  df-id 5464  df-xp 5566  df-rel 5567  df-cnv 5568  df-co 5569  df-dm 5570  df-rn 5571  df-res 5572  df-ima 5573  df-iota 6347  df-fun 6391  df-fn 6392  df-f 6393  df-fv 6397  df-ov 7225  df-oprab 7226  df-mpo 7227  df-1st 7770  df-2nd 7771  df-map 8519
This theorem is referenced by:  mapssfset  8541  fsetex  8546
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