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Theorem fnmptf 5983
Description: The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.) (Revised by Thierry Arnoux, 10-May-2017.)
Hypothesis
Ref Expression
mptfnf.0 𝑥𝐴
Assertion
Ref Expression
fnmptf (∀𝑥𝐴 𝐵𝑉 → (𝑥𝐴𝐵) Fn 𝐴)

Proof of Theorem fnmptf
StepHypRef Expression
1 elex 3202 . . 3 (𝐵𝑉𝐵 ∈ V)
21ralimi 2948 . 2 (∀𝑥𝐴 𝐵𝑉 → ∀𝑥𝐴 𝐵 ∈ V)
3 mptfnf.0 . . 3 𝑥𝐴
43mptfnf 5982 . 2 (∀𝑥𝐴 𝐵 ∈ V ↔ (𝑥𝐴𝐵) Fn 𝐴)
52, 4sylib 208 1 (∀𝑥𝐴 𝐵𝑉 → (𝑥𝐴𝐵) Fn 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  wnfc 2748  wral 2908  Vcvv 3190  cmpt 4683   Fn wfn 5852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-fun 5859  df-fn 5860
This theorem is referenced by:  offval2f  6874  esumgsum  29930  esumc  29936  bj-mptval  32746  rfovcnvf1od  37819  dssmapf1od  37836  ntrrn  37941  dssmapntrcls  37947
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