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Theorem mptfnd 41875
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 416 . . . 4 (𝜑 → (𝑥𝐴𝐵𝑉))
4 elex 3462 . . . 4 (𝐵𝑉𝐵 ∈ V)
53, 4syl6 35 . . 3 (𝜑 → (𝑥𝐴𝐵 ∈ V))
61, 5ralrimi 3183 . 2 (𝜑 → ∀𝑥𝐴 𝐵 ∈ V)
7 mptfnd.1 . . 3 𝑥𝐴
87mptfnf 6459 . 2 (∀𝑥𝐴 𝐵 ∈ V ↔ (𝑥𝐴𝐵) Fn 𝐴)
96, 8sylib 221 1 (𝜑 → (𝑥𝐴𝐵) Fn 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wnf 1785  wcel 2112  wnfc 2939  wral 3109  Vcvv 3444  cmpt 5113   Fn wfn 6323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-fun 6330  df-fn 6331
This theorem is referenced by:  smflimsuplem2  43449
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