MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnexd Structured version   Visualization version   GIF version

Theorem fnexd 6980
Description: If the domain of a function is a set, the function is a set. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
fnexd.1 (𝜑𝐹 Fn 𝐴)
fnexd.2 (𝜑𝐴𝑉)
Assertion
Ref Expression
fnexd (𝜑𝐹 ∈ V)

Proof of Theorem fnexd
StepHypRef Expression
1 fnexd.1 . 2 (𝜑𝐹 Fn 𝐴)
2 fnexd.2 . 2 (𝜑𝐴𝑉)
3 fnex 6979 . 2 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐹 ∈ V)
41, 2, 3syl2anc 586 1 (𝜑𝐹 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2110  Vcvv 3494   Fn wfn 6349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pr 5329
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  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:  suppcofnd  7870  limsupequzlem  42003
  Copyright terms: Public domain W3C validator