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Theorem mpt2fun 6165
 Description: The maps-to notation for an operation is always a function. (Contributed by Scott Fenton, 21-Mar-2012.)
Hypothesis
Ref Expression
mpt2fun.1
Assertion
Ref Expression
mpt2fun
Distinct variable group:   ,
Allowed substitution hints:   (,)   (,)   (,)   (,)

Proof of Theorem mpt2fun
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqtr3 2455 . . . . . 6
21ad2ant2l 727 . . . . 5
32gen2 1556 . . . 4
4 eqeq1 2442 . . . . . 6
54anbi2d 685 . . . . 5
65mo4 2314 . . . 4
73, 6mpbir 201 . . 3
87funoprab 6163 . 2
9 mpt2fun.1 . . . 4
10 df-mpt2 6079 . . . 4
119, 10eqtri 2456 . . 3
1211funeqi 5467 . 2
138, 12mpbir 201 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359  wal 1549   wceq 1652   wcel 1725  wmo 2282   wfun 5441  coprab 6075   cmpt2 6076 This theorem is referenced by:  ofexg  6302  mpt2exxg  6415  imasvscafn  13755  coapm  14219  oppglsm  15269  xkococnlem  17684  ucnima  18304  ucnprima  18305  fmucnd  18315  tpr2rico  24303  elunirnmbfm  24596  aovmpt4g  28033 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pr 4396 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-br 4206  df-opab 4260  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-fun 5449  df-oprab 6078  df-mpt2 6079
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