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Theorem mptfng 5024
 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 2712 . . 3 (𝐵 ∈ V ↔ ∃!𝑦 𝑦 = 𝐵)
21ralbii 2330 . 2 (∀𝑥𝐴 𝐵 ∈ V ↔ ∀𝑥𝐴 ∃!𝑦 𝑦 = 𝐵)
3 mptfng.1 . . . 4 𝐹 = (𝑥𝐴𝐵)
4 df-mpt 3820 . . . 4 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
53, 4eqtri 2060 . . 3 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
65fnopabg 5022 . 2 (∀𝑥𝐴 ∃!𝑦 𝑦 = 𝐵𝐹 Fn 𝐴)
72, 6bitri 173 1 (∀𝑥𝐴 𝐵 ∈ V ↔ 𝐹 Fn 𝐴)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393  ∃!weu 1900  ∀wral 2306  Vcvv 2557  {copab 3817   ↦ cmpt 3818   Fn wfn 4897 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-fun 4904  df-fn 4905 This theorem is referenced by:  fnmpt  5025  fnmpti  5027  mpteqb  5261
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