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Theorem fexd 41264
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 6986 . 2 ((𝐹:𝐴𝐵𝐴𝐶) → 𝐹 ∈ V)
41, 2, 3syl2anc 584 1 (𝜑𝐹 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  Vcvv 3500  wf 6350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362
This theorem is referenced by:  limsupval3  41857  limsuppnfdlem  41866  limsupvaluz  41873  limsuppnflem  41875  limsupre2lem  41889  climuzlem  41908  climisp  41911  climxrrelem  41914  climxrre  41915  liminfval5  41930  limsupgtlem  41942  liminfvalxr  41948  liminflelimsupuz  41950  liminfgelimsupuz  41953  liminflimsupclim  41972  liminflbuz2  41980  xlimclim2lem  42004  climxlim2  42011  nsssmfmbflem  42939
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