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Theorem fnmptd 6482
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 413 . . 3 (𝜑 → (𝑥𝐴𝐵𝑉))
41, 3ralrimi 3213 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝑉)
5 fnmptd.3 . . 3 𝐹 = (𝑥𝐴𝐵)
65fnmpt 6481 . 2 (∀𝑥𝐴 𝐵𝑉𝐹 Fn 𝐴)
74, 6syl 17 1 (𝜑𝐹 Fn 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wnf 1775  wcel 2105  wral 3135  cmpt 5137   Fn wfn 6343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-fun 6350  df-fn 6351
This theorem is referenced by:  limsupequzmptlem  41885  liminfval2  41925  smflimmpt  42961  smflimsuplem7  42977
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