Theorem fvco2 5962

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 5947	. . . . . 6 |- ( ( G Fn A /\ X e. A ) -> { ( G ` X ) } = ( G " { X } ) )
2	1	imaeq2d 5186	. . . . 5 |- ( ( G Fn A /\ X e. A ) -> ( F " { ( G ` X ) } ) = ( F " ( G " { X } ) ) )
3		imaco 5358	. . . . 5 |- ( ( F o. G ) " { X } ) = ( F " ( G " { X } ) )
4	2, 3	syl6reqr 2514	. . . 4 |- ( ( G Fn A /\ X e. A ) -> ( ( F o. G ) " { X } ) = ( F " { ( G ` X ) } ) )
5	4	eleq2d 2524	. . 3 |- ( ( G Fn A /\ X e. A ) -> ( x e. ( ( F o. G ) " { X } ) <-> x e. ( F " { ( G ` X ) } ) ) )
6	5	iotabidv 5585	. 2 |- ( ( G Fn A /\ X e. A ) -> ( iota x x e. ( ( F o. G ) " { X } ) ) = ( iota x x e. ( F " { ( G ` X ) } ) ) )
7		dffv3 5883	. 2 |- ( ( F o. G ) ` X ) = ( iota x x e. ( ( F o. G ) " { X } ) )
8		dffv3 5883	. 2 |- ( F ` ( G ` X ) ) = ( iota x x e. ( F " { ( G ` X ) } ) )
9	6, 7, 8	3eqtr4g 2520	1 |- ( ( G Fn A /\ X e. A ) -> ( ( F o. G ) ` X ) = ( F ` ( G ` X ) ) )

Syntax hints:   -> wi 4   /\ wa 375   = wceq 1454   e. wcel 1897   {csn 3979   "cima 4855   o. ccom 4856   iotacio 5562   Fn wfn 5595   ` cfv 5600

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-br 4416  df-opab 4475  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5564  df-fun 5602  df-fn 5603  df-fv 5608

This theorem is referenced by:  fvco  5963  fvco3  5964  fvco4i  5965  fvcofneq  6052  ofco  6577  curry1  6914  curry2  6917  enfixsn  7706  smobeth  9036  fpwwe  9096  addpqnq  9388  mulpqnq  9391  revco  12967  ccatco  12968  cshco  12969  swrdco  12970  isoval  15718  prdsidlem  16616  gsumwmhm  16677  prdsinvlem  16842  gsmsymgrfixlem1  17116  f1omvdconj  17135  pmtrfinv  17150  symggen  17159  symgtrinv  17161  pmtr3ncomlem1  17162  ringidval  17785  prdsmgp  17886  lmhmco  18314  evlslem1  18786  evlsvar  18794  chrrhm  19150  zrhcofipsgn  19209  dsmmbas2  19348  dsmm0cl  19351  frlmbas  19366  frlmup3  19406  frlmup4  19407  f1lindf  19428  lindfmm  19433  m1detdiag  19670  1stccnp  20525  prdstopn  20691  xpstopnlem2  20874  uniioombllem6  22594  0vfval  26273  cnre2csqlem  28764  mblfinlem2  32022  rabren3dioph  35702  hausgraph  36133  stoweidlem59  37957  afvco2  38715