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Theorem fvco2 5185
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  X  F  o.  G `  X  F `  G `  X

Proof of Theorem fvco2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 fnsnfv 5175 . . . . . 6  G  Fn  X  { G `  X }  G " { X }
21imaeq2d 4611 . . . . 5  G  Fn  X  F " { G `  X }  F " G
" { X }
3 imaco 4769 . . . . 5  F  o.  G
" { X }  F " G " { X }
42, 3syl6reqr 2088 . . . 4  G  Fn  X  F  o.  G " { X }  F " { G `  X }
54eleq2d 2104 . . 3  G  Fn  X  F  o.  G " { X }  F
" { G `
 X }
65iotabidv 4831 . 2  G  Fn  X  iota  F  o.  G " { X }  iota  F " { G `  X }
7 dffv3g 5117 . . 3  X  F  o.  G `  X  iota  F  o.  G " { X }
87adantl 262 . 2  G  Fn  X  F  o.  G `  X  iota  F  o.  G " { X }
9 funfvex 5135 . . . 4  Fun  G  X  dom  G  G `  X  _V
109funfni 4942 . . 3  G  Fn  X  G `  X  _V
11 dffv3g 5117 . . 3  G `  X  _V  F `  G `  X  iota  F " { G `  X }
1210, 11syl 14 . 2  G  Fn  X  F `  G `  X  iota  F " { G `
 X }
136, 8, 123eqtr4d 2079 1  G  Fn  X  F  o.  G `  X  F `  G `  X
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242   wcel 1390   _Vcvv 2551   {csn 3367   "cima 4291    o. ccom 4292   iotacio 4808    Fn wfn 4840   ` cfv 4845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853
This theorem is referenced by:  fvco  5186  fvco3  5187  ofco  5671
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