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Theorem fvco3d 6743
Description: Value of a function composition. Deduction form of fvco3 6742. (Contributed by Stanislas Polu, 9-Mar-2020.)
Hypotheses
Ref Expression
fvco3d.1 (𝜑𝐺:𝐴𝐵)
fvco3d.2 (𝜑𝐶𝐴)
Assertion
Ref Expression
fvco3d (𝜑 → ((𝐹𝐺)‘𝐶) = (𝐹‘(𝐺𝐶)))

Proof of Theorem fvco3d
StepHypRef Expression
1 fvco3d.1 . 2 (𝜑𝐺:𝐴𝐵)
2 fvco3d.2 . 2 (𝜑𝐶𝐴)
3 fvco3 6742 . 2 ((𝐺:𝐴𝐵𝐶𝐴) → ((𝐹𝐺)‘𝐶) = (𝐹‘(𝐺𝐶)))
41, 2, 3syl2anc 587 1 (𝜑 → ((𝐹𝐺)‘𝐶) = (𝐹‘(𝐺𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2114  ccom 5536  wf 6330  cfv 6334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-fv 6342
This theorem is referenced by:  suppcoss  7858  yonedainv  17522  frgpcyg  20263  comet  23118  pmtrcnel  30764  selvval2lem4  39377  extoimad  40801  imo72b2lem0  40802  imo72b2lem1  40807
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