Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fvcod Structured version   Visualization version   GIF version

Theorem fvcod 39942
Description: Value of a function composition. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
fvcod.g (𝜑 → Fun 𝐺)
fvcod.a (𝜑𝐴 ∈ dom 𝐺)
fvcod.h 𝐻 = (𝐹𝐺)
Assertion
Ref Expression
fvcod (𝜑 → (𝐻𝐴) = (𝐹‘(𝐺𝐴)))

Proof of Theorem fvcod
StepHypRef Expression
1 fvcod.h . . . 4 𝐻 = (𝐹𝐺)
21fveq1i 6332 . . 3 (𝐻𝐴) = ((𝐹𝐺)‘𝐴)
32a1i 11 . 2 (𝜑 → (𝐻𝐴) = ((𝐹𝐺)‘𝐴))
4 fvcod.g . . 3 (𝜑 → Fun 𝐺)
5 fvcod.a . . 3 (𝜑𝐴 ∈ dom 𝐺)
6 fvco 6415 . . 3 ((Fun 𝐺𝐴 ∈ dom 𝐺) → ((𝐹𝐺)‘𝐴) = (𝐹‘(𝐺𝐴)))
74, 5, 6syl2anc 693 . 2 (𝜑 → ((𝐹𝐺)‘𝐴) = (𝐹‘(𝐺𝐴)))
83, 7eqtrd 2803 1 (𝜑 → (𝐻𝐴) = (𝐹‘(𝐺𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1629  wcel 2143  dom cdm 5248  ccom 5252  Fun wfun 6024  cfv 6030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1868  ax-4 1883  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2145  ax-9 2152  ax-10 2172  ax-11 2188  ax-12 2201  ax-13 2406  ax-ext 2749  ax-sep 4911  ax-nul 4919  ax-pow 4970  ax-pr 5033
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1071  df-tru 1632  df-ex 1851  df-nf 1856  df-sb 2048  df-eu 2620  df-mo 2621  df-clab 2756  df-cleq 2762  df-clel 2765  df-nfc 2900  df-ne 2942  df-ral 3064  df-rex 3065  df-rab 3068  df-v 3350  df-sbc 3585  df-dif 3723  df-un 3725  df-in 3727  df-ss 3734  df-nul 4061  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4572  df-br 4784  df-opab 4844  df-id 5156  df-xp 5254  df-rel 5255  df-cnv 5256  df-co 5257  df-dm 5258  df-rn 5259  df-res 5260  df-ima 5261  df-iota 5993  df-fun 6032  df-fn 6033  df-fv 6038
This theorem is referenced by:  subsaliuncllem  41093
  Copyright terms: Public domain W3C validator