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Theorem mptfnd 44619
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 411 . . . 4 (𝜑 → (𝑥𝐴𝐵𝑉))
4 elex 3490 . . . 4 (𝐵𝑉𝐵 ∈ V)
53, 4syl6 35 . . 3 (𝜑 → (𝑥𝐴𝐵 ∈ V))
61, 5ralrimi 3250 . 2 (𝜑 → ∀𝑥𝐴 𝐵 ∈ V)
7 mptfnd.1 . . 3 𝑥𝐴
87mptfnf 6693 . 2 (∀𝑥𝐴 𝐵 ∈ V ↔ (𝑥𝐴𝐵) Fn 𝐴)
96, 8sylib 217 1 (𝜑 → (𝑥𝐴𝐵) Fn 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wnf 1777  wcel 2098  wnfc 2878  wral 3057  Vcvv 3471  cmpt 5233   Fn wfn 6546
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-fun 6553  df-fn 6554
This theorem is referenced by:  smflimsuplem2  46211
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