Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  mpt2fun Unicode version

Theorem mpt2fun 5631
 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 2075 . . . . . 6
21ad2ant2l 485 . . . . 5
32gen2 1355 . . . 4
4 eqeq1 2062 . . . . . 6
54anbi2d 445 . . . . 5
65mo4 1977 . . . 4
73, 6mpbir 138 . . 3
87funoprab 5629 . 2
9 mpt2fun.1 . . . 4
10 df-mpt2 5545 . . . 4
119, 10eqtri 2076 . . 3
1211funeqi 4950 . 2
138, 12mpbir 138 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 101  wal 1257   wceq 1259   wcel 1409  wmo 1917   wfun 4924  coprab 5541   cmpt2 5542 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-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-fun 4932  df-oprab 5544  df-mpt2 5545 This theorem is referenced by:  elmpt2cl  5726  ofexg  5744  mpt2exxg  5861  mpt2xopn0yelv  5885
 Copyright terms: Public domain W3C validator