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Theorem funmpt 5129
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 5128 . 2 Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
2 df-mpt 3959 . . 3 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
32funeqi 5112 . 2 (Fun (𝑥𝐴𝐵) ↔ Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)})
41, 3mpbir 145 1 Fun (𝑥𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1314  wcel 1463  {copab 3956  cmpt 3957  Fun wfun 5085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-fun 5093
This theorem is referenced by:  funmpt2  5130  fmptco  5552  resfunexg  5607  mptexg  5611  brtpos2  6114  tposfun  6123  rdgtfr  6237  rdgruledefgg  6238  rdgon  6249  freccllem  6265  frecfcllem  6267  hashinfom  10464  hashennn  10466  negfi  10939  tgrest  12233  dvrecap  12720
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