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Theorem fvco 7006
Description: Value of a function composition. Similar to Exercise 5 of [TakeutiZaring] p. 28. (Contributed by NM, 22-Apr-2006.) (Proof shortened by Mario Carneiro, 26-Dec-2014.)
Assertion
Ref Expression
fvco ((Fun 𝐺𝐴 ∈ dom 𝐺) → ((𝐹𝐺)‘𝐴) = (𝐹‘(𝐺𝐴)))

Proof of Theorem fvco
StepHypRef Expression
1 funfn 6597 . 2 (Fun 𝐺𝐺 Fn dom 𝐺)
2 fvco2 7005 . 2 ((𝐺 Fn dom 𝐺𝐴 ∈ dom 𝐺) → ((𝐹𝐺)‘𝐴) = (𝐹‘(𝐺𝐴)))
31, 2sylanb 581 1 ((Fun 𝐺𝐴 ∈ dom 𝐺) → ((𝐹𝐺)‘𝐴) = (𝐹‘(𝐺𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  dom cdm 5688  ccom 5692  Fun wfun 6556   Fn wfn 6557  cfv 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-fv 6570
This theorem is referenced by:  fin23lem30  10379  hashkf  14367  hashgval  14368  gsumpropd2lem  18704  ofco2  22472  opfv  32660  xppreima  32661  psgnfzto1stlem  33102  cycpmfv1  33115  cycpmfv2  33116  cyc3co2  33142  smatlem  33757  mdetpmtr1  33783  madjusmdetlem2  33788  madjusmdetlem4  33790  eulerpartlemgvv  34357  eulerpartlemgu  34358  sseqfv2  34375  reprpmtf1o  34619  hgt750lemg  34647  aks5lem2  42168  comptiunov2i  43695  choicefi  45142  fvcod  45169  evthiccabs  45448  cncficcgt0  45843  dvsinax  45868  fvvolioof  45944  fvvolicof  45946  stirlinglem14  46042  fourierdlem42  46104  hoicvr  46503  hoi2toco  46562  ovolval3  46602  ovolval4lem1  46604  ovnovollem1  46611  ovnovollem2  46612
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