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Theorem mapfset 8875
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 8861 (which does not require ax-rep 5289) 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 8880), 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 3477 . . . 4 𝑚 ∈ V
2 feq1 6708 . . . 4 (𝑓 = 𝑚 → (𝑓:𝐴𝐵𝑚:𝐴𝐵))
31, 2elab 3669 . . 3 (𝑚 ∈ {𝑓𝑓:𝐴𝐵} ↔ 𝑚:𝐴𝐵)
4 simpr 483 . . . . . . 7 ((𝑚:𝐴𝐵𝐵𝑉) → 𝐵𝑉)
5 dmfex 7919 . . . . . . . . 9 ((𝑚 ∈ V ∧ 𝑚:𝐴𝐵) → 𝐴 ∈ V)
61, 5mpan 688 . . . . . . . 8 (𝑚:𝐴𝐵𝐴 ∈ V)
76adantr 479 . . . . . . 7 ((𝑚:𝐴𝐵𝐵𝑉) → 𝐴 ∈ V)
84, 7elmapd 8865 . . . . . 6 ((𝑚:𝐴𝐵𝐵𝑉) → (𝑚 ∈ (𝐵m 𝐴) ↔ 𝑚:𝐴𝐵))
98exbiri 809 . . . . 5 (𝑚:𝐴𝐵 → (𝐵𝑉 → (𝑚:𝐴𝐵𝑚 ∈ (𝐵m 𝐴))))
109pm2.43b 55 . . . 4 (𝐵𝑉 → (𝑚:𝐴𝐵𝑚 ∈ (𝐵m 𝐴)))
11 elmapi 8874 . . . 4 (𝑚 ∈ (𝐵m 𝐴) → 𝑚:𝐴𝐵)
1210, 11impbid1 224 . . 3 (𝐵𝑉 → (𝑚:𝐴𝐵𝑚 ∈ (𝐵m 𝐴)))
133, 12bitrid 282 . 2 (𝐵𝑉 → (𝑚 ∈ {𝑓𝑓:𝐴𝐵} ↔ 𝑚 ∈ (𝐵m 𝐴)))
1413eqrdv 2726 1 (𝐵𝑉 → {𝑓𝑓:𝐴𝐵} = (𝐵m 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = 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-csb 3895  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-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  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-1st 7999  df-2nd 8000  df-map 8853
This theorem is referenced by:  mapssfset  8876  fsetex  8881
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