Theorem fsetex 8849

Description: The set of functions between two classes exists if the codomain exists. Generalization of mapex 7933 to arbitrary domains. (Contributed by AV, 14-Aug-2024.)

Assertion

Ref	Expression
fsetex	⊢ (𝐵 ∈ 𝑉 → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V)

Distinct variable groups:   𝐴,𝑓   𝐵,𝑓

Allowed substitution hint:   𝑉(𝑓)

Proof of Theorem fsetex

Step	Hyp	Ref	Expression
1		mapfset 8843	. 2 ⊢ (𝐵 ∈ 𝑉 → {𝑓 ∣ 𝑓:𝐴⟶𝐵} = (𝐵 ↑m 𝐴))
2		ovex 7441	. 2 ⊢ (𝐵 ↑m 𝐴) ∈ V
3	1, 2	eqeltrdi 2877	1 ⊢ (𝐵 ∈ 𝑉 → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2149  {cab 2747  Vcvv 3463  ⟶wf 6529  (class class class)co 7408   ↑_m cmap 8820

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7982  df-2nd 7983  df-map 8822

This theorem is referenced by:  f1setex  8850  fsetcdmex  8856  fsetexb  8857  isghm  19282  sn-isghm  43290