Theorem fvco2 5933

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.

Step	Hyp	Ref	Expression
1		fnsnfv 5918	. . . . . 6 |- ( ( G Fn A /\ X e. A ) -> { ( G ` X ) } = ( G " { X } ) )
2	1	imaeq2d 5327	. . . . 5 |- ( ( G Fn A /\ X e. A ) -> ( F " { ( G ` X ) } ) = ( F " ( G " { X } ) ) )
3		imaco 5502	. . . . 5 |- ( ( F o. G ) " { X } ) = ( F " ( G " { X } ) )
4	2, 3	syl6reqr 2503	. . . 4 |- ( ( G Fn A /\ X e. A ) -> ( ( F o. G ) " { X } ) = ( F " { ( G ` X ) } ) )
5	4	eleq2d 2513	. . 3 |- ( ( G Fn A /\ X e. A ) -> ( x e. ( ( F o. G ) " { X } ) <-> x e. ( F " { ( G ` X ) } ) ) )
6	5	iotabidv 5562	. 2 |- ( ( G Fn A /\ X e. A ) -> ( iota x x e. ( ( F o. G ) " { X } ) ) = ( iota x x e. ( F " { ( G ` X ) } ) ) )
7		dffv3 5852	. 2 |- ( ( F o. G ) ` X ) = ( iota x x e. ( ( F o. G ) " { X } ) )
8		dffv3 5852	. 2 |- ( F ` ( G ` X ) ) = ( iota x x e. ( F " { ( G ` X ) } ) )
9	6, 7, 8	3eqtr4g 2509	1 |- ( ( G Fn A /\ X e. A ) -> ( ( F o. G ) ` X ) = ( F ` ( G ` X ) ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1383   e. wcel 1804   {csn 4014   "cima 4992   o. ccom 4993   iotacio 5539   Fn wfn 5573   ` cfv 5578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-fv 5586

This theorem is referenced by:  fvco  5934  fvco3  5935  fvco4i  5936  fvcofneq  6024  ofco  6545  curry1  6877  curry2  6880  enfixsn  7628  smobeth  8964  fpwwe  9027  addpqnq  9319  mulpqnq  9322  revco  12782  ccatco  12783  cshco  12784  swrdco  12785  isoval  15141  prdsidlem  15931  gsumwmhm  15992  prdsinvlem  16157  gsmsymgrfixlem1  16431  f1omvdconj  16450  pmtrfinv  16465  symggen  16474  symgtrinv  16476  pmtr3ncomlem1  16477  ringidval  17134  prdsmgp  17238  lmhmco  17668  evlslem1  18163  evlsvar  18171  chrrhm  18546  zrhcofipsgn  18607  dsmmbas2  18746  dsmm0cl  18749  frlmbas  18764  frlmbasOLD  18765  frlmup3  18812  frlmup4  18813  f1lindf  18835  lindfmm  18840  m1detdiag  19077  1stccnp  19941  prdstopn  20107  xpstopnlem2  20290  uniioombllem6  21975  0vfval  25477  cnre2csqlem  27870  mblfinlem2  30028  rabren3dioph  30725  hausgraph  31148  stoweidlem59  31795  afvco2  32215