Theorem mapfset 8769

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 8755 (which does not require ax-rep 5215) 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 8774), 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 3438	. . . 4 ⊢ 𝑚 ∈ V
2		feq1 6625	. . . 4 ⊢ (𝑓 = 𝑚 → (𝑓:𝐴⟶𝐵 ↔ 𝑚:𝐴⟶𝐵))
3	1, 2	elab 3633	. . 3 ⊢ (𝑚 ∈ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ↔ 𝑚:𝐴⟶𝐵)
4		simpr 484	. . . . . . 7 ⊢ ((𝑚:𝐴⟶𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉)
5		dmfex 7830	. . . . . . . . 9 ⊢ ((𝑚 ∈ V ∧ 𝑚:𝐴⟶𝐵) → 𝐴 ∈ V)
6	1, 5	mpan 690	. . . . . . . 8 ⊢ (𝑚:𝐴⟶𝐵 → 𝐴 ∈ V)
7	6	adantr 480	. . . . . . 7 ⊢ ((𝑚:𝐴⟶𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ V)
8	4, 7	elmapd 8759	. . . . . 6 ⊢ ((𝑚:𝐴⟶𝐵 ∧ 𝐵 ∈ 𝑉) → (𝑚 ∈ (𝐵 ↑m 𝐴) ↔ 𝑚:𝐴⟶𝐵))
9	8	exbiri 810	. . . . 5 ⊢ (𝑚:𝐴⟶𝐵 → (𝐵 ∈ 𝑉 → (𝑚:𝐴⟶𝐵 → 𝑚 ∈ (𝐵 ↑m 𝐴))))
10	9	pm2.43b 55	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝑚:𝐴⟶𝐵 → 𝑚 ∈ (𝐵 ↑m 𝐴)))
11		elmapi 8768	. . . 4 ⊢ (𝑚 ∈ (𝐵 ↑m 𝐴) → 𝑚:𝐴⟶𝐵)
12	10, 11	impbid1 225	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝑚:𝐴⟶𝐵 ↔ 𝑚 ∈ (𝐵 ↑m 𝐴)))
13	3, 12	bitrid 283	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝑚 ∈ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ↔ 𝑚 ∈ (𝐵 ↑m 𝐴)))
14	13	eqrdv 2728	1 ⊢ (𝐵 ∈ 𝑉 → {𝑓 ∣ 𝑓:𝐴⟶𝐵} = (𝐵 ↑m 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2110  {cab 2708  Vcvv 3434  ⟶wf 6473  (class class class)co 7341   ↑_m cmap 8745

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-fv 6485  df-ov 7344  df-oprab 7345  df-mpo 7346  df-1st 7916  df-2nd 7917  df-map 8747

This theorem is referenced by:  mapssfset  8770  fsetex  8775