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Theorem fexd 39793
Description: If the domain of a mapping is a set, the function is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
fexd.1 (𝜑𝐹:𝐴𝐵)
fexd.2 (𝜑𝐴𝐶)
Assertion
Ref Expression
fexd (𝜑𝐹 ∈ V)

Proof of Theorem fexd
StepHypRef Expression
1 fexd.1 . 2 (𝜑𝐹:𝐴𝐵)
2 fexd.2 . 2 (𝜑𝐴𝐶)
3 fex 6651 . 2 ((𝐹:𝐴𝐵𝐴𝐶) → 𝐹 ∈ V)
41, 2, 3syl2anc 696 1 (𝜑𝐹 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2137  Vcvv 3338  wf 6043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pr 5053
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-reu 3055  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055
This theorem is referenced by:  limsupval3  40425  limsuppnfdlem  40434  limsupvaluz  40441  limsuppnflem  40443  limsupre2lem  40457  climuzlem  40476  climisp  40479  climxrrelem  40482  climxrre  40483  liminfval5  40498  limsupgtlem  40510  liminfvalxr  40516  liminflelimsupuz  40518  liminfgelimsupuz  40521  liminflimsupclim  40540  xlimclim2lem  40566  climxlim2  40573  nsssmfmbflem  41490
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