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Theorem fvco2 6943
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 𝐴𝑋𝐴) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))

Proof of Theorem fvco2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 imaco 6208 . . . . 5 ((𝐹𝐺) “ {𝑋}) = (𝐹 “ (𝐺 “ {𝑋}))
2 fnsnfv 6925 . . . . . 6 ((𝐺 Fn 𝐴𝑋𝐴) → {(𝐺𝑋)} = (𝐺 “ {𝑋}))
32imaeq2d 6018 . . . . 5 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐹 “ {(𝐺𝑋)}) = (𝐹 “ (𝐺 “ {𝑋})))
41, 3eqtr4id 2796 . . . 4 ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺) “ {𝑋}) = (𝐹 “ {(𝐺𝑋)}))
54eleq2d 2824 . . 3 ((𝐺 Fn 𝐴𝑋𝐴) → (𝑥 ∈ ((𝐹𝐺) “ {𝑋}) ↔ 𝑥 ∈ (𝐹 “ {(𝐺𝑋)})))
65iotabidv 6485 . 2 ((𝐺 Fn 𝐴𝑋𝐴) → (℩𝑥𝑥 ∈ ((𝐹𝐺) “ {𝑋})) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺𝑋)})))
7 dffv3 6843 . 2 ((𝐹𝐺)‘𝑋) = (℩𝑥𝑥 ∈ ((𝐹𝐺) “ {𝑋}))
8 dffv3 6843 . 2 (𝐹‘(𝐺𝑋)) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺𝑋)}))
96, 7, 83eqtr4g 2802 1 ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  {csn 4591  cima 5641  ccom 5642  cio 6451   Fn wfn 6496  cfv 6501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-fv 6509
This theorem is referenced by:  fvco  6944  fvco3  6945  fvco4i  6947  fvcofneq  7048  ofco  7645  curry1  8041  curry2  8044  fsplitfpar  8055  enfixsn  9032  updjudhcoinlf  9875  updjudhcoinrg  9876  updjud  9877  smobeth  10529  fpwwe  10589  addpqnq  10881  mulpqnq  10884  revco  14730  ccatco  14731  cshco  14732  swrdco  14733  isoval  17655  prdsidlem  18595  gsumwmhm  18662  prdsinvlem  18863  gsmsymgrfixlem1  19216  f1omvdconj  19235  pmtrfinv  19250  symggen  19259  symgtrinv  19261  pmtr3ncomlem1  19262  ringidval  19922  prdsmgp  20041  lmhmco  20520  chrrhm  20950  cofipsgn  21013  dsmmbas2  21159  dsmm0cl  21162  frlmbas  21177  frlmup3  21222  frlmup4  21223  f1lindf  21244  lindfmm  21249  evlslem1  21508  evlsvar  21516  m1detdiag  21962  1stccnp  22829  prdstopn  22995  xpstopnlem2  23178  uniioombllem6  24968  ex-fpar  29448  0vfval  29590  ghmquskerco  32236  cnre2csqlem  32531  mblfinlem2  36145  rabren3dioph  41167  hausgraph  41568  stoweidlem59  44374  afvco2  45482  isomushgr  46092  isomgrtrlem  46104  ackvalsucsucval  46848
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