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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 416 . . 3 (𝜑 → (𝑥𝐴𝐵𝑉))
41, 3ralrimi 3129 . 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 399   = wceq 1542  wnf 1790  wcel 2114  wral 3054  cmpt 5120   Fn wfn 6344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ral 3059  df-v 3402  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-fun 6351  df-fn 6352
This theorem is referenced by:  nsgmgc  31181  nsgqusf1o  31185  metakunt33  39788  limsupequzmptlem  42851  liminfval2  42891  smflimmpt  43922  smflimsuplem7  43938  cfsetsnfsetfo  44133
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