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Theorem fmpti 5353
Description: Functionality of the mapping operation. (Contributed by NM, 19-Mar-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
Hypotheses
Ref Expression
fmpt.1  |-  F  =  ( x  e.  A  |->  C )
fmpti.2  |-  ( x  e.  A  ->  C  e.  B )
Assertion
Ref Expression
fmpti  |-  F : A
--> B
Distinct variable groups:    x, A    x, B
Allowed substitution hints:    C( x)    F( x)

Proof of Theorem fmpti
StepHypRef Expression
1 fmpti.2 . . 3  |-  ( x  e.  A  ->  C  e.  B )
21rgen 2417 . 2  |-  A. x  e.  A  C  e.  B
3 fmpt.1 . . 3  |-  F  =  ( x  e.  A  |->  C )
43fmpt 5351 . 2  |-  ( A. x  e.  A  C  e.  B  <->  F : A --> B )
52, 4mpbi 143 1  |-  F : A
--> B
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    e. wcel 1434   A.wral 2349    |-> cmpt 3847   -->wf 4928
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-fv 4940
This theorem is referenced by:  cjf  9872  ref  9880  imf  9881  absf  10134
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