Theorem fvco2 5923

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 5908	. . . . . 6 |- ( ( G Fn A /\ X e. A ) -> { ( G ` X ) } = ( G " { X } ) )
2	1	imaeq2d 5325	. . . . 5 |- ( ( G Fn A /\ X e. A ) -> ( F " { ( G ` X ) } ) = ( F " ( G " { X } ) ) )
3		imaco 5495	. . . . 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 5555	. 2 |- ( ( G Fn A /\ X e. A ) -> ( iota x x e. ( ( F o. G ) " { X } ) ) = ( iota x x e. ( F " { ( G ` X ) } ) ) )
7		dffv3 5844	. 2 |- ( ( F o. G ) ` X ) = ( iota x x e. ( ( F o. G ) " { X } ) )
8		dffv3 5844	. 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 367   = wceq 1398   e. wcel 1823   {csn 4016   "cima 4991   o. ccom 4992   iotacio 5532   Fn wfn 5565   ` cfv 5570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-fv 5578

This theorem is referenced by:  fvco  5924  fvco3  5925  fvco4i  5926  fvcofneq  6015  ofco  6533  curry1  6865  curry2  6868  enfixsn  7619  smobeth  8952  fpwwe  9013  addpqnq  9305  mulpqnq  9308  revco  12794  ccatco  12795  cshco  12796  swrdco  12797  isoval  15256  prdsidlem  16154  gsumwmhm  16215  prdsinvlem  16380  gsmsymgrfixlem1  16654  f1omvdconj  16673  pmtrfinv  16688  symggen  16697  symgtrinv  16699  pmtr3ncomlem1  16700  ringidval  17353  prdsmgp  17457  lmhmco  17887  evlslem1  18382  evlsvar  18390  chrrhm  18746  zrhcofipsgn  18805  dsmmbas2  18944  dsmm0cl  18947  frlmbas  18962  frlmbasOLD  18963  frlmup3  19005  frlmup4  19006  f1lindf  19027  lindfmm  19032  m1detdiag  19269  1stccnp  20132  prdstopn  20298  xpstopnlem2  20481  uniioombllem6  22166  0vfval  25700  cnre2csqlem  28130  mblfinlem2  30295  rabren3dioph  30991  hausgraph  31416  stoweidlem59  32083  afvco2  32503