Theorem fvco2 6533

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 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋)))

Proof of Theorem fvco2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnsnfv 6518	. . . . . 6 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → {(𝐺‘𝑋)} = (𝐺 “ {𝑋}))
2	1	imaeq2d 5720	. . . . 5 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐹 “ {(𝐺‘𝑋)}) = (𝐹 “ (𝐺 “ {𝑋})))
3		imaco 5894	. . . . 5 ⊢ ((𝐹 ∘ 𝐺) “ {𝑋}) = (𝐹 “ (𝐺 “ {𝑋}))
4	2, 3	syl6reqr 2832	. . . 4 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺) “ {𝑋}) = (𝐹 “ {(𝐺‘𝑋)}))
5	4	eleq2d 2844	. . 3 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑥 ∈ ((𝐹 ∘ 𝐺) “ {𝑋}) ↔ 𝑥 ∈ (𝐹 “ {(𝐺‘𝑋)})))
6	5	iotabidv 6120	. 2 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (℩𝑥𝑥 ∈ ((𝐹 ∘ 𝐺) “ {𝑋})) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺‘𝑋)})))
7		dffv3 6442	. 2 ⊢ ((𝐹 ∘ 𝐺)‘𝑋) = (℩𝑥𝑥 ∈ ((𝐹 ∘ 𝐺) “ {𝑋}))
8		dffv3 6442	. 2 ⊢ (𝐹‘(𝐺‘𝑋)) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺‘𝑋)}))
9	6, 7, 8	3eqtr4g 2838	1 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋)))

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2106  {csn 4397   “ cima 5358   ∘ ccom 5359  ℩cio 6097   Fn wfn 6130  ‘cfv 6135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-fv 6143

This theorem is referenced by:  fvco  6534  fvco3  6535  fvco4i  6536  fvcofneq  6631  ofco  7194  curry1  7550  curry2  7553  enfixsn  8357  updjudhcoinlf  9091  updjudhcoinrg  9092  updjud  9093  smobeth  9743  fpwwe  9803  addpqnq  10095  mulpqnq  10098  revco  13985  ccatco  13986  cshco  13987  swrdco  13988  isoval  16810  prdsidlem  17708  gsumwmhm  17769  prdsinvlem  17911  gsmsymgrfixlem1  18230  f1omvdconj  18249  pmtrfinv  18264  symggen  18273  symgtrinv  18275  pmtr3ncomlem1  18276  ringidval  18890  prdsmgp  18997  lmhmco  19438  evlslem1  19911  evlsvar  19919  chrrhm  20275  cofipsgn  20334  zrhcofipsgnOLD  20335  dsmmbas2  20480  dsmm0cl  20483  frlmbas  20498  frlmup3  20543  frlmup4  20544  f1lindf  20565  lindfmm  20570  m1detdiag  20808  1stccnp  21674  prdstopn  21840  xpstopnlem2  22023  uniioombllem6  23792  0vfval  28033  cnre2csqlem  30554  mblfinlem2  34057  rabren3dioph  38321  hausgraph  38731  stoweidlem59  41185  afvco2  42199  isomushgr  42721  isomgrtrlem  42733