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

Theorem fnmptd 6643
Description: The maps-to notation defines a function with domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
fnmptd.1 𝑥𝜑
fnmptd.2 ((𝜑𝑥𝐴) → 𝐵𝑉)
fnmptd.3 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
fnmptd (𝜑𝐹 Fn 𝐴)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem fnmptd
StepHypRef Expression
1 fnmptd.1 . . 3 𝑥𝜑
2 fnmptd.2 . . . 4 ((𝜑𝑥𝐴) → 𝐵𝑉)
32ex 414 . . 3 (𝜑 → (𝑥𝐴𝐵𝑉))
41, 3ralrimi 3239 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝑉)
5 fnmptd.3 . . 3 𝐹 = (𝑥𝐴𝐵)
65fnmpt 6642 . 2 (∀𝑥𝐴 𝐵𝑉𝐹 Fn 𝐴)
74, 6syl 17 1 (𝜑𝐹 Fn 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wnf 1786  wcel 2107  wral 3061  cmpt 5189   Fn wfn 6492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-fun 6499  df-fn 6500
This theorem is referenced by:  nsgmgc  32238  nsgqusf1o  32242  ply1gsumz  32339  ply1degltdimlem  32374  ply1annidllem  32426  metakunt33  40655  limsupequzmptlem  44055  liminfval2  44095  smflimmpt  45137  smflimsuplem7  45153  cfsetsnfsetfo  45380
  Copyright terms: Public domain W3C validator