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Theorem fsetex 8875
Description: The set of functions between two classes exists if the codomain exists. Generalization of mapex 8851 to arbitrary domains. (Contributed by AV, 14-Aug-2024.)
Assertion
Ref Expression
fsetex (𝐵𝑉 → {𝑓𝑓:𝐴𝐵} ∈ V)
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓
Allowed substitution hint:   𝑉(𝑓)

Proof of Theorem fsetex
StepHypRef Expression
1 mapfset 8869 . 2 (𝐵𝑉 → {𝑓𝑓:𝐴𝐵} = (𝐵m 𝐴))
2 ovex 7453 . 2 (𝐵m 𝐴) ∈ V
31, 2eqeltrdi 2837 1 (𝐵𝑉 → {𝑓𝑓:𝐴𝐵} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2099  {cab 2705  Vcvv 3471  wf 6544  (class class class)co 7420  m cmap 8845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 7993  df-2nd 7994  df-map 8847
This theorem is referenced by:  f1setex  8876  fsetcdmex  8882  fsetexb  8883
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