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Theorem fvmptg 5280
Description: Value of a function given in maps-to notation. (Contributed by NM, 2-Oct-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypotheses
Ref Expression
fvmptg.1  |-  ( x  =  A  ->  B  =  C )
fvmptg.2  |-  F  =  ( x  e.  D  |->  B )
Assertion
Ref Expression
fvmptg  |-  ( ( A  e.  D  /\  C  e.  R )  ->  ( F `  A
)  =  C )
Distinct variable groups:    x, A    x, C    x, D
Allowed substitution hints:    B( x)    R( x)    F( x)

Proof of Theorem fvmptg
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eqid 2082 . 2  |-  C  =  C
2 fvmptg.1 . . . 4  |-  ( x  =  A  ->  B  =  C )
32eqeq2d 2093 . . 3  |-  ( x  =  A  ->  (
y  =  B  <->  y  =  C ) )
4 eqeq1 2088 . . 3  |-  ( y  =  C  ->  (
y  =  C  <->  C  =  C ) )
5 moeq 2768 . . . 4  |-  E* y 
y  =  B
65a1i 9 . . 3  |-  ( x  e.  D  ->  E* y  y  =  B
)
7 fvmptg.2 . . . 4  |-  F  =  ( x  e.  D  |->  B )
8 df-mpt 3849 . . . 4  |-  ( x  e.  D  |->  B )  =  { <. x ,  y >.  |  ( x  e.  D  /\  y  =  B ) }
97, 8eqtri 2102 . . 3  |-  F  =  { <. x ,  y
>.  |  ( x  e.  D  /\  y  =  B ) }
103, 4, 6, 9fvopab3ig 5278 . 2  |-  ( ( A  e.  D  /\  C  e.  R )  ->  ( C  =  C  ->  ( F `  A )  =  C ) )
111, 10mpi 15 1  |-  ( ( A  e.  D  /\  C  e.  R )  ->  ( F `  A
)  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   E*wmo 1943   {copab 3846    |-> cmpt 3847   ` cfv 4932
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-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-iota 4897  df-fun 4934  df-fv 4940
This theorem is referenced by:  fvmpt  5281  fvmpts  5282  fvmpt3  5283  fvmpt2  5286  f1mpt  5442  fnofval  5752  caofinvl  5764  1stvalg  5800  2ndvalg  5801  brtpos2  5900  rdgon  6035  frec0g  6046  freccllem  6051  frecfcllem  6053  frecsuclem  6055  sucinc  6089  sucinc2  6090  omcl  6105  oeicl  6106  oav2  6107  omv2  6109  cardval3ex  6513  ceilqval  9388  frec2uzzd  9482  frec2uzsucd  9483  monoord2  9552  iseqdistr  9567  serile  9571  sizeinf  9802  sizeennn  9804  cjval  9870  reval  9874  imval  9875  cvg1nlemcau  10008  cvg1nlemres  10009  absval  10025  resqrexlemglsq  10046  resqrexlemga  10047  climmpt  10277  climle  10310  climcvg1nlem  10324  isumrblem  10337
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