Theorem fvco2 5942

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 5927	. . . . . 6 |- ( ( G Fn A /\ X e. A ) -> { ( G ` X ) } = ( G " { X } ) )
2	1	imaeq2d 5337	. . . . 5 |- ( ( G Fn A /\ X e. A ) -> ( F " { ( G ` X ) } ) = ( F " ( G " { X } ) ) )
3		imaco 5512	. . . . 5 |- ( ( F o. G ) " { X } ) = ( F " ( G " { X } ) )
4	2, 3	syl6reqr 2527	. . . 4 |- ( ( G Fn A /\ X e. A ) -> ( ( F o. G ) " { X } ) = ( F " { ( G ` X ) } ) )
5	4	eleq2d 2537	. . 3 |- ( ( G Fn A /\ X e. A ) -> ( x e. ( ( F o. G ) " { X } ) <-> x e. ( F " { ( G ` X ) } ) ) )
6	5	iotabidv 5572	. 2 |- ( ( G Fn A /\ X e. A ) -> ( iota x x e. ( ( F o. G ) " { X } ) ) = ( iota x x e. ( F " { ( G ` X ) } ) ) )
7		dffv3 5862	. 2 |- ( ( F o. G ) ` X ) = ( iota x x e. ( ( F o. G ) " { X } ) )
8		dffv3 5862	. 2 |- ( F ` ( G ` X ) ) = ( iota x x e. ( F " { ( G ` X ) } ) )
9	6, 7, 8	3eqtr4g 2533	1 |- ( ( G Fn A /\ X e. A ) -> ( ( F o. G ) ` X ) = ( F ` ( G ` X ) ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1379   e. wcel 1767   {csn 4027   "cima 5002   o. ccom 5003   iotacio 5549   Fn wfn 5583   ` cfv 5588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-fv 5596

This theorem is referenced by:  fvco  5943  fvco3  5944  fvco4i  5945  fvcofneq  6029  ofco  6544  curry1  6875  curry2  6878  enfixsn  7626  smobeth  8961  fpwwe  9024  addpqnq  9316  mulpqnq  9319  revco  12763  ccatco  12764  cshco  12765  swrdco  12766  isoval  15020  prdsidlem  15771  gsumwmhm  15845  prdsinvlem  15988  gsmsymgrfixlem1  16257  f1omvdconj  16277  pmtrfinv  16292  symggen  16301  symgtrinv  16303  pmtr3ncomlem1  16304  rngidval  16957  prdsmgp  17060  lmhmco  17489  evlslem1  17983  evlsvar  17991  chrrhm  18363  zrhcofipsgn  18424  dsmmbas2  18563  dsmm0cl  18566  frlmbas  18581  frlmbasOLD  18582  frlmup3  18629  frlmup4  18630  f1lindf  18652  lindfmm  18657  m1detdiag  18894  1stccnp  19757  prdstopn  19892  xpstopnlem2  20075  uniioombllem6  21760  0vfval  25203  cnre2csqlem  27556  mblfinlem2  29657  rabren3dioph  30381  hausgraph  30805  stoweidlem59  31387  afvco2  31756