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Theorem funmpt 4881
Description: A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.)
Assertion
Ref Expression
funmpt Fun (x AB)

Proof of Theorem funmpt
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 funopab4 4880 . 2 Fun {⟨x, y⟩ ∣ (x A y = B)}
2 df-mpt 3811 . . 3 (x AB) = {⟨x, y⟩ ∣ (x A y = B)}
32funeqi 4865 . 2 (Fun (x AB) ↔ Fun {⟨x, y⟩ ∣ (x A y = B)})
41, 3mpbir 134 1 Fun (x AB)
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242   wcel 1390  {copab 3808  cmpt 3809  Fun wfun 4839
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-fun 4847
This theorem is referenced by:  funmpt2  4882  fmptco  5273  resfunexg  5325  mptexg  5329  brtpos2  5807  tposfun  5816  rdgtfr  5901  rdgruledefgg  5902
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