Theorem fvco2 5787

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 5772	. . . . . 6 |- ( ( G Fn A /\ X e. A ) -> { ( G ` X ) } = ( G " { X } ) )
2	1	imaeq2d 5190	. . . . 5 |- ( ( G Fn A /\ X e. A ) -> ( F " { ( G ` X ) } ) = ( F " ( G " { X } ) ) )
3		imaco 5364	. . . . 5 |- ( ( F o. G ) " { X } ) = ( F " ( G " { X } ) )
4	2, 3	syl6reqr 2494	. . . 4 |- ( ( G Fn A /\ X e. A ) -> ( ( F o. G ) " { X } ) = ( F " { ( G ` X ) } ) )
5	4	eleq2d 2510	. . 3 |- ( ( G Fn A /\ X e. A ) -> ( x e. ( ( F o. G ) " { X } ) <-> x e. ( F " { ( G ` X ) } ) ) )
6	5	iotabidv 5423	. 2 |- ( ( G Fn A /\ X e. A ) -> ( iota x x e. ( ( F o. G ) " { X } ) ) = ( iota x x e. ( F " { ( G ` X ) } ) ) )
7		dffv3 5708	. 2 |- ( ( F o. G ) ` X ) = ( iota x x e. ( ( F o. G ) " { X } ) )
8		dffv3 5708	. 2 |- ( F ` ( G ` X ) ) = ( iota x x e. ( F " { ( G ` X ) } ) )
9	6, 7, 8	3eqtr4g 2500	1 |- ( ( G Fn A /\ X e. A ) -> ( ( F o. G ) ` X ) = ( F ` ( G ` X ) ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1369   e. wcel 1756   {csn 3898   "cima 4864   o. ccom 4865   iotacio 5400   Fn wfn 5434   ` cfv 5439

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-fv 5447

This theorem is referenced by:  fvco  5788  fvco3  5789  fvco4i  5790  fvcofneq  5872  ofco  6361  curry1  6685  curry2  6688  enfixsn  7441  smobeth  8771  fpwwe  8834  addpqnq  9128  mulpqnq  9131  revco  12483  ccatco  12484  cshco  12485  swrdco  12486  isoval  14724  prdsidlem  15474  gsumwmhm  15544  prdsinvlem  15684  gsmsymgrfixlem1  15953  f1omvdconj  15973  pmtrfinv  15988  symggen  15997  symgtrinv  15999  pmtr3ncomlem1  16000  rngidval  16627  prdsmgp  16724  lmhmco  17146  evlslem1  17623  evlsvar  17631  chrrhm  17984  zrhcofipsgn  18045  dsmmbas2  18184  dsmm0cl  18187  frlmbas  18202  frlmbasOLD  18203  frlmup3  18250  frlmup4  18251  f1lindf  18273  lindfmm  18278  1stccnp  19088  prdstopn  19223  xpstopnlem2  19406  uniioombllem6  21090  0vfval  24006  cnre2csqlem  26362  mblfinlem2  28455  rabren3dioph  29180  hausgraph  29606  stoweidlem59  29880  afvco2  30108  m1detdiag  30931