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Theorem mptfnd 44430
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 3485 . . . 4 (𝐵𝑉𝐵 ∈ V)
53, 4syl6 35 . . 3 (𝜑 → (𝑥𝐴𝐵 ∈ V))
61, 5ralrimi 3246 . 2 (𝜑 → ∀𝑥𝐴 𝐵 ∈ V)
7 mptfnd.1 . . 3 𝑥𝐴
87mptfnf 6675 . 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 395  wnf 1777  wcel 2098  wnfc 2875  wral 3053  Vcvv 3466  cmpt 5221   Fn wfn 6528
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 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-fun 6535  df-fn 6536
This theorem is referenced by:  smflimsuplem2  46022
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