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Theorem mapfset 8791
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 8778 (which does not require ax-rep 5243) 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 8796), 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 3448 . . . 4 𝑚 ∈ V
2 feq1 6650 . . . 4 (𝑓 = 𝑚 → (𝑓:𝐴𝐵𝑚:𝐴𝐵))
31, 2elab 3631 . . 3 (𝑚 ∈ {𝑓𝑓:𝐴𝐵} ↔ 𝑚:𝐴𝐵)
4 simpr 486 . . . . . . 7 ((𝑚:𝐴𝐵𝐵𝑉) → 𝐵𝑉)
5 dmfex 7845 . . . . . . . . 9 ((𝑚 ∈ V ∧ 𝑚:𝐴𝐵) → 𝐴 ∈ V)
61, 5mpan 689 . . . . . . . 8 (𝑚:𝐴𝐵𝐴 ∈ V)
76adantr 482 . . . . . . 7 ((𝑚:𝐴𝐵𝐵𝑉) → 𝐴 ∈ V)
84, 7elmapd 8782 . . . . . 6 ((𝑚:𝐴𝐵𝐵𝑉) → (𝑚 ∈ (𝐵m 𝐴) ↔ 𝑚:𝐴𝐵))
98exbiri 810 . . . . 5 (𝑚:𝐴𝐵 → (𝐵𝑉 → (𝑚:𝐴𝐵𝑚 ∈ (𝐵m 𝐴))))
109pm2.43b 55 . . . 4 (𝐵𝑉 → (𝑚:𝐴𝐵𝑚 ∈ (𝐵m 𝐴)))
11 elmapi 8790 . . . 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 2731 1 (𝐵𝑉 → {𝑓𝑓:𝐴𝐵} = (𝐵m 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  {cab 2710  Vcvv 3444  wf 6493  (class class class)co 7358  m cmap 8768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-map 8770
This theorem is referenced by:  mapssfset  8792  fsetex  8797
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