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Theorem fnmptd 6488
 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 3221 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝑉)
5 fnmptd.3 . . 3 𝐹 = (𝑥𝐴𝐵)
65fnmpt 6487 . 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 1530  Ⅎwnf 1777   ∈ wcel 2107  ∀wral 3143   ↦ cmpt 5143   Fn wfn 6349 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-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-ral 3148  df-rab 3152  df-v 3502  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-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-fun 6356  df-fn 6357 This theorem is referenced by:  limsupequzmptlem  41893  liminfval2  41933  smflimmpt  42969  smflimsuplem7  42985
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