Theorem mapfset 6935

Description: If 𝐵 is a set, the value of the set exponentiation (𝐵 ↑_𝑚 𝐴) is the class of all functions from 𝐴 to 𝐵. Generalisation of mapvalg 6922 to arbitrary domains. Note that the class {𝑓 ∣ 𝑓:𝐴⟶𝐵} 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 6940), 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	⊢ (𝐵 ∈ 𝑉 → {𝑓 ∣ 𝑓:𝐴⟶𝐵} = (𝐵 ↑𝑚 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.

Step	Hyp	Ref	Expression
1		vex 2824	. . . 4 ⊢ 𝑚 ∈ V
2		feq1 5511	. . . 4 ⊢ (𝑓 = 𝑚 → (𝑓:𝐴⟶𝐵 ↔ 𝑚:𝐴⟶𝐵))
3	1, 2	elab 2970	. . 3 ⊢ (𝑚 ∈ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ↔ 𝑚:𝐴⟶𝐵)
4		simpr 110	. . . . . . 7 ⊢ ((𝑚:𝐴⟶𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉)
5		dmfex 5577	. . . . . . . . 9 ⊢ ((𝑚 ∈ V ∧ 𝑚:𝐴⟶𝐵) → 𝐴 ∈ V)
6	1, 5	mpan 428	. . . . . . . 8 ⊢ (𝑚:𝐴⟶𝐵 → 𝐴 ∈ V)
7	6	adantr 276	. . . . . . 7 ⊢ ((𝑚:𝐴⟶𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ V)
8	4, 7	elmapd 6926	. . . . . 6 ⊢ ((𝑚:𝐴⟶𝐵 ∧ 𝐵 ∈ 𝑉) → (𝑚 ∈ (𝐵 ↑𝑚 𝐴) ↔ 𝑚:𝐴⟶𝐵))
9	8	exbiri 382	. . . . 5 ⊢ (𝑚:𝐴⟶𝐵 → (𝐵 ∈ 𝑉 → (𝑚:𝐴⟶𝐵 → 𝑚 ∈ (𝐵 ↑𝑚 𝐴))))
10	9	pm2.43b 52	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝑚:𝐴⟶𝐵 → 𝑚 ∈ (𝐵 ↑𝑚 𝐴)))
11		elmapi 6934	. . . 4 ⊢ (𝑚 ∈ (𝐵 ↑𝑚 𝐴) → 𝑚:𝐴⟶𝐵)
12	10, 11	impbid1 142	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝑚:𝐴⟶𝐵 ↔ 𝑚 ∈ (𝐵 ↑𝑚 𝐴)))
13	3, 12	bitrid 192	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝑚 ∈ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ↔ 𝑚 ∈ (𝐵 ↑𝑚 𝐴)))
14	13	eqrdv 2236	1 ⊢ (𝐵 ∈ 𝑉 → {𝑓 ∣ 𝑓:𝐴⟶𝐵} = (𝐵 ↑𝑚 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  {cab 2224  Vcvv 2821  ⟶wf 5368  (class class class)co 6075   ↑_𝑚 cmap 6912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-map 6914

This theorem is referenced by:  mapssfsetg  6936