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Theorem mptfng 5125
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 2784 . . 3 (𝐵 ∈ V ↔ ∃!𝑦 𝑦 = 𝐵)
21ralbii 2384 . 2 (∀𝑥𝐴 𝐵 ∈ V ↔ ∀𝑥𝐴 ∃!𝑦 𝑦 = 𝐵)
3 mptfng.1 . . . 4 𝐹 = (𝑥𝐴𝐵)
4 df-mpt 3893 . . . 4 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
53, 4eqtri 2108 . . 3 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
65fnopabg 5123 . 2 (∀𝑥𝐴 ∃!𝑦 𝑦 = 𝐵𝐹 Fn 𝐴)
72, 6bitri 182 1 (∀𝑥𝐴 𝐵 ∈ V ↔ 𝐹 Fn 𝐴)
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103   = wceq 1289  wcel 1438  ∃!weu 1948  wral 2359  Vcvv 2619  {copab 3890  cmpt 3891   Fn wfn 4997
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-fun 5004  df-fn 5005
This theorem is referenced by:  fnmpt  5126  fnmpti  5128  mpteqb  5377
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