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Theorem mpt2fun 6404
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:   ,

Proof of Theorem mpt2fun
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqtr3 2485 . . . . . 6
21ad2ant2l 745 . . . . 5
32gen2 1619 . . . 4
4 eqeq1 2461 . . . . . 6
54anbi2d 703 . . . . 5
65mo4 2337 . . . 4
73, 6mpbir 209 . . 3
87funoprab 6402 . 2
9 mpt2fun.1 . . . 4
10 df-mpt2 6301 . . . 4
119, 10eqtri 2486 . . 3
1211funeqi 5613 . 2
138, 12mpbir 209 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  /\wa 369  A.wal 1393  =wceq 1395  e.wcel 1818  E*wmo 2283  Funwfun 5587  {coprab 6297  e.cmpt2 6298
This theorem is referenced by:  ofexg  6544  mpt2exxg  6874  mpt2curryd  7017  imasvscafn  14934  coapm  15398  oppglsm  16662  gsum2d2lem  17001  evlslem2  18180  xkococnlem  20160  ucnima  20784  ucnprima  20785  fmucnd  20795  txomap  27837  tpr2rico  27894  elunirnmbfm  28224  aovmpt4g  32286  mpt2exxg2  32927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-fun 5595  df-oprab 6300  df-mpt2 6301
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