Theorem fvco2 6847

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		imaco 6144	. . . . 5 ⊢ ((𝐹 ∘ 𝐺) “ {𝑋}) = (𝐹 “ (𝐺 “ {𝑋}))
2		fnsnfv 6829	. . . . . 6 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → {(𝐺‘𝑋)} = (𝐺 “ {𝑋}))
3	2	imaeq2d 5958	. . . . 5 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐹 “ {(𝐺‘𝑋)}) = (𝐹 “ (𝐺 “ {𝑋})))
4	1, 3	eqtr4id 2798	. . . 4 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺) “ {𝑋}) = (𝐹 “ {(𝐺‘𝑋)}))
5	4	eleq2d 2824	. . 3 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑥 ∈ ((𝐹 ∘ 𝐺) “ {𝑋}) ↔ 𝑥 ∈ (𝐹 “ {(𝐺‘𝑋)})))
6	5	iotabidv 6402	. 2 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (℩𝑥𝑥 ∈ ((𝐹 ∘ 𝐺) “ {𝑋})) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺‘𝑋)})))
7		dffv3 6752	. 2 ⊢ ((𝐹 ∘ 𝐺)‘𝑋) = (℩𝑥𝑥 ∈ ((𝐹 ∘ 𝐺) “ {𝑋}))
8		dffv3 6752	. 2 ⊢ (𝐹‘(𝐺‘𝑋)) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺‘𝑋)}))
9	6, 7, 8	3eqtr4g 2804	1 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {csn 4558   “ cima 5583   ∘ ccom 5584  ℩cio 6374   Fn wfn 6413  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-fv 6426

This theorem is referenced by:  fvco  6848  fvco3  6849  fvco4i  6851  fvcofneq  6951  ofco  7534  curry1  7915  curry2  7918  fsplitfpar  7930  enfixsn  8821  updjudhcoinlf  9621  updjudhcoinrg  9622  updjud  9623  smobeth  10273  fpwwe  10333  addpqnq  10625  mulpqnq  10628  revco  14475  ccatco  14476  cshco  14477  swrdco  14478  isoval  17394  prdsidlem  18332  gsumwmhm  18399  prdsinvlem  18599  gsmsymgrfixlem1  18950  f1omvdconj  18969  pmtrfinv  18984  symggen  18993  symgtrinv  18995  pmtr3ncomlem1  18996  ringidval  19654  prdsmgp  19764  lmhmco  20220  chrrhm  20647  cofipsgn  20710  dsmmbas2  20854  dsmm0cl  20857  frlmbas  20872  frlmup3  20917  frlmup4  20918  f1lindf  20939  lindfmm  20944  evlslem1  21202  evlsvar  21210  m1detdiag  21654  1stccnp  22521  prdstopn  22687  xpstopnlem2  22870  uniioombllem6  24657  ex-fpar  28727  0vfval  28869  cnre2csqlem  31762  mblfinlem2  35742  rabren3dioph  40553  hausgraph  40953  stoweidlem59  43490  afvco2  44555  isomushgr  45166  isomgrtrlem  45178  ackvalsucsucval  45922