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Theorem fvco2 5456
Description: Value of a function composition. Similar to second part of Theorem 3H of [Enderton] p. 47. (Contributed by NM, 9-Oct-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by Stefan O'Rear, 16-Oct-2014.)
Assertion
Ref Expression
fvco2  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( ( F  o.  G ) `  X
)  =  ( F `
 ( G `  X ) ) )

Proof of Theorem fvco2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fnsnfv 5446 . . . . . 6  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  { ( G `  X ) }  =  ( G " { X } ) )
21imaeq2d 4849 . . . . 5  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( F " {
( G `  X
) } )  =  ( F " ( G " { X }
) ) )
3 imaco 5012 . . . . 5  |-  ( ( F  o.  G )
" { X }
)  =  ( F
" ( G " { X } ) )
42, 3syl6reqr 2167 . . . 4  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( ( F  o.  G ) " { X } )  =  ( F " { ( G `  X ) } ) )
54eleq2d 2185 . . 3  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( x  e.  ( ( F  o.  G
) " { X } )  <->  x  e.  ( F " { ( G `  X ) } ) ) )
65iotabidv 5077 . 2  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( iota x x  e.  ( ( F  o.  G ) " { X } ) )  =  ( iota x x  e.  ( F " { ( G `  X ) } ) ) )
7 dffv3g 5383 . . 3  |-  ( X  e.  A  ->  (
( F  o.  G
) `  X )  =  ( iota x x  e.  ( ( F  o.  G ) " { X } ) ) )
87adantl 273 . 2  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( ( F  o.  G ) `  X
)  =  ( iota
x x  e.  ( ( F  o.  G
) " { X } ) ) )
9 funfvex 5404 . . . 4  |-  ( ( Fun  G  /\  X  e.  dom  G )  -> 
( G `  X
)  e.  _V )
109funfni 5191 . . 3  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( G `  X
)  e.  _V )
11 dffv3g 5383 . . 3  |-  ( ( G `  X )  e.  _V  ->  ( F `  ( G `  X ) )  =  ( iota x x  e.  ( F " { ( G `  X ) } ) ) )
1210, 11syl 14 . 2  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( F `  ( G `  X )
)  =  ( iota
x x  e.  ( F " { ( G `  X ) } ) ) )
136, 8, 123eqtr4d 2158 1  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( ( F  o.  G ) `  X
)  =  ( F `
 ( G `  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314    e. wcel 1463   _Vcvv 2658   {csn 3495   "cima 4510    o. ccom 4511   iotacio 5054    Fn wfn 5086   ` cfv 5091
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-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-fv 5099
This theorem is referenced by:  fvco  5457  fvco3  5458  ofco  5966  updjudhcoinlf  6931  updjudhcoinrg  6932  updjud  6933  caseinl  6942  caseinr  6943  ctm  6960  enomnilem  6976
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