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Theorem funmpt 4966
Description: A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.)
Assertion
Ref Expression
funmpt Fun (𝑥𝐴𝐵)

Proof of Theorem funmpt
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 funopab4 4965 . 2 Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
2 df-mpt 3848 . . 3 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
32funeqi 4950 . 2 (Fun (𝑥𝐴𝐵) ↔ Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)})
41, 3mpbir 138 1 Fun (𝑥𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wa 101   = wceq 1259  wcel 1409  {copab 3845  cmpt 3846  Fun wfun 4924
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-fun 4932
This theorem is referenced by:  funmpt2  4967  fmptco  5358  resfunexg  5410  mptexg  5414  brtpos2  5897  tposfun  5906  rdgtfr  5992  rdgruledefgg  5993
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