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Theorem mptfng 5918
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 3344 . . 3 (𝐵 ∈ V ↔ ∃!𝑦 𝑦 = 𝐵)
21ralbii 2962 . 2 (∀𝑥𝐴 𝐵 ∈ V ↔ ∀𝑥𝐴 ∃!𝑦 𝑦 = 𝐵)
3 mptfng.1 . . . 4 𝐹 = (𝑥𝐴𝐵)
4 df-mpt 4639 . . . 4 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
53, 4eqtri 2631 . . 3 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
65fnopabg 5916 . 2 (∀𝑥𝐴 ∃!𝑦 𝑦 = 𝐵𝐹 Fn 𝐴)
72, 6bitri 262 1 (∀𝑥𝐴 𝐵 ∈ V ↔ 𝐹 Fn 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382   = wceq 1474  wcel 1976  ∃!weu 2457  wral 2895  Vcvv 3172  {copab 4636  cmpt 4637   Fn wfn 5785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-fun 5792  df-fn 5793
This theorem is referenced by:  fnmpt  5919  fnmpti  5921  mpteqb  6192  ofmpteq  6791  bdayfo  30880  fobigcup  30983  dihf11lem  35369
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