Theorem fvco2 6751

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 6736	. . . . . 6 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → {(𝐺‘𝑋)} = (𝐺 “ {𝑋}))
2	1	imaeq2d 5922	. . . . 5 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐹 “ {(𝐺‘𝑋)}) = (𝐹 “ (𝐺 “ {𝑋})))
3		imaco 6097	. . . . 5 ⊢ ((𝐹 ∘ 𝐺) “ {𝑋}) = (𝐹 “ (𝐺 “ {𝑋}))
4	2, 3	syl6reqr 2873	. . . 4 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺) “ {𝑋}) = (𝐹 “ {(𝐺‘𝑋)}))
5	4	eleq2d 2896	. . 3 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑥 ∈ ((𝐹 ∘ 𝐺) “ {𝑋}) ↔ 𝑥 ∈ (𝐹 “ {(𝐺‘𝑋)})))
6	5	iotabidv 6332	. 2 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (℩𝑥𝑥 ∈ ((𝐹 ∘ 𝐺) “ {𝑋})) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺‘𝑋)})))
7		dffv3 6659	. 2 ⊢ ((𝐹 ∘ 𝐺)‘𝑋) = (℩𝑥𝑥 ∈ ((𝐹 ∘ 𝐺) “ {𝑋}))
8		dffv3 6659	. 2 ⊢ (𝐹‘(𝐺‘𝑋)) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺‘𝑋)}))
9	6, 7, 8	3eqtr4g 2879	1 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋)))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1531   ∈ wcel 2108  {csn 4559   “ cima 5551   ∘ ccom 5552  ℩cio 6305   Fn wfn 6343  ‘cfv 6348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356

This theorem is referenced by:  fvco  6752  fvco3  6753  fvco4i  6755  fvcofneq  6852  ofco  7421  curry1  7791  curry2  7794  fsplitfpar  7806  enfixsn  8618  updjudhcoinlf  9353  updjudhcoinrg  9354  updjud  9355  smobeth  10000  fpwwe  10060  addpqnq  10352  mulpqnq  10355  revco  14188  ccatco  14189  cshco  14190  swrdco  14191  isoval  17027  prdsidlem  17935  gsumwmhm  18002  prdsinvlem  18200  gsmsymgrfixlem1  18547  f1omvdconj  18566  pmtrfinv  18581  symggen  18590  symgtrinv  18592  pmtr3ncomlem1  18593  ringidval  19245  prdsmgp  19352  lmhmco  19807  evlslem1  20287  evlsvar  20295  chrrhm  20670  cofipsgn  20729  dsmmbas2  20873  dsmm0cl  20876  frlmbas  20891  frlmup3  20936  frlmup4  20937  f1lindf  20958  lindfmm  20963  m1detdiag  21198  1stccnp  22062  prdstopn  22228  xpstopnlem2  22411  uniioombllem6  24181  ex-fpar  28233  0vfval  28375  cnre2csqlem  31146  mblfinlem2  34922  rabren3dioph  39403  hausgraph  39803  stoweidlem59  42335  afvco2  43366  isomushgr  43982  isomgrtrlem  43994