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Theorem mptfnd 45248
Description: The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.) (Revised by Thierry Arnoux, 10-May-2017.)
Hypotheses
Ref Expression
mptfnd.1 𝑥𝐴
mptfnd.2 𝑥𝜑
mptfnd.3 ((𝜑𝑥𝐴) → 𝐵𝑉)
Assertion
Ref Expression
mptfnd (𝜑 → (𝑥𝐴𝐵) Fn 𝐴)

Proof of Theorem mptfnd
StepHypRef Expression
1 mptfnd.2 . . 3 𝑥𝜑
2 mptfnd.3 . . . . 5 ((𝜑𝑥𝐴) → 𝐵𝑉)
32ex 412 . . . 4 (𝜑 → (𝑥𝐴𝐵𝑉))
4 elex 3501 . . . 4 (𝐵𝑉𝐵 ∈ V)
53, 4syl6 35 . . 3 (𝜑 → (𝑥𝐴𝐵 ∈ V))
61, 5ralrimi 3257 . 2 (𝜑 → ∀𝑥𝐴 𝐵 ∈ V)
7 mptfnd.1 . . 3 𝑥𝐴
87mptfnf 6703 . 2 (∀𝑥𝐴 𝐵 ∈ V ↔ (𝑥𝐴𝐵) Fn 𝐴)
96, 8sylib 218 1 (𝜑 → (𝑥𝐴𝐵) Fn 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wnf 1783  wcel 2108  wnfc 2890  wral 3061  Vcvv 3480  cmpt 5225   Fn wfn 6556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-fun 6563  df-fn 6564
This theorem is referenced by:  smflimsuplem2  46836
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