Theorem mapfset 8831

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 8817 (which does not require ax-rep 5228) 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 8836), 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 3459	. . . 4 ⊢ 𝑚 ∈ V
2		feq1 6669	. . . 4 ⊢ (𝑓 = 𝑚 → (𝑓:𝐴⟶𝐵 ↔ 𝑚:𝐴⟶𝐵))
3	1, 2	elab 3639	. . 3 ⊢ (𝑚 ∈ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ↔ 𝑚:𝐴⟶𝐵)
4		simpr 488	. . . . . . 7 ⊢ ((𝑚:𝐴⟶𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉)
5		dmfex 7886	. . . . . . . . 9 ⊢ ((𝑚 ∈ V ∧ 𝑚:𝐴⟶𝐵) → 𝐴 ∈ V)
6	1, 5	mpan 700	. . . . . . . 8 ⊢ (𝑚:𝐴⟶𝐵 → 𝐴 ∈ V)
7	6	adantr 484	. . . . . . 7 ⊢ ((𝑚:𝐴⟶𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ V)
8	4, 7	elmapd 8821	. . . . . 6 ⊢ ((𝑚:𝐴⟶𝐵 ∧ 𝐵 ∈ 𝑉) → (𝑚 ∈ (𝐵 ↑m 𝐴) ↔ 𝑚:𝐴⟶𝐵))
9	8	exbiri 820	. . . . 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 8830	. . . 4 ⊢ (𝑚 ∈ (𝐵 ↑m 𝐴) → 𝑚:𝐴⟶𝐵)
12	10, 11	impbid1 227	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝑚:𝐴⟶𝐵 ↔ 𝑚 ∈ (𝐵 ↑m 𝐴)))
13	3, 12	bitrid 285	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝑚 ∈ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ↔ 𝑚 ∈ (𝐵 ↑m 𝐴)))
14	13	eqrdv 2761	1 ⊢ (𝐵 ∈ 𝑉 → {𝑓 ∣ 𝑓:𝐴⟶𝐵} = (𝐵 ↑m 𝐴))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1561   ∈ wcel 2143  {cab 2741  Vcvv 3455  ⟶wf 6517  (class class class)co 7396   ↑_m cmap 8808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971  df-map 8810

This theorem is referenced by:  mapssfset  8832  fsetex  8837