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Theorem mpofun 5839
 Description: The maps-to notation for an operation is always a function. (Contributed by Scott Fenton, 21-Mar-2012.)
Hypothesis
Ref Expression
mpofun.1
Assertion
Ref Expression
mpofun
Distinct variable group:   ,
Allowed substitution hints:   (,)   (,)   (,)   (,)

Proof of Theorem mpofun
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqtr3 2135 . . . . . 6
21ad2ant2l 497 . . . . 5
32gen2 1409 . . . 4
4 eqeq1 2122 . . . . . 6
54anbi2d 457 . . . . 5
65mo4 2036 . . . 4
73, 6mpbir 145 . . 3
87funoprab 5837 . 2
9 mpofun.1 . . . 4
10 df-mpo 5745 . . . 4
119, 10eqtri 2136 . . 3
1211funeqi 5112 . 2
138, 12mpbir 145 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103  wal 1312   wceq 1314   wcel 1463  wmo 1976   wfun 5085  coprab 5741   cmpo 5742 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-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-fun 5093  df-oprab 5744  df-mpo 5745 This theorem is referenced by:  elmpocl  5934  ofexg  5952  mpoexxg  6074  mpoxopn0yelv  6102
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