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Theorem mptfng 5932
 Description: The maps-to notation defines a function with domain. (Contributed by Scott Fenton, 21-Mar-2011.)
Hypothesis
Ref Expression
mptfng.1 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
mptfng (∀𝑥𝐴 𝐵 ∈ V ↔ 𝐹 Fn 𝐴)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem mptfng
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eueq 3345 . . 3 (𝐵 ∈ V ↔ ∃!𝑦 𝑦 = 𝐵)
21ralbii 2963 . 2 (∀𝑥𝐴 𝐵 ∈ V ↔ ∀𝑥𝐴 ∃!𝑦 𝑦 = 𝐵)
3 mptfng.1 . . . 4 𝐹 = (𝑥𝐴𝐵)
4 df-mpt 4645 . . . 4 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
53, 4eqtri 2632 . . 3 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
65fnopabg 5930 . 2 (∀𝑥𝐴 ∃!𝑦 𝑦 = 𝐵𝐹 Fn 𝐴)
72, 6bitri 263 1 (∀𝑥𝐴 𝐵 ∈ V ↔ 𝐹 Fn 𝐴)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃!weu 2458  ∀wral 2896  Vcvv 3173  {copab 4642   ↦ cmpt 4643   Fn wfn 5799 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-fun 5806  df-fn 5807 This theorem is referenced by:  fnmpt  5933  fnmpti  5935  mpteqb  6207  ofmpteq  6814  bdayfo  31074  fobigcup  31177  dihf11lem  35573
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