Mathbox for Stanislas Polu < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  wfximgfd Structured version   Visualization version   GIF version

Theorem wfximgfd 40800
 Description: The value of a function on its domain is in the image of the function. (Contributed by Stanislas Polu, 9-Mar-2020.)
Hypotheses
Ref Expression
wfximgfd.1 (𝜑𝐶𝐴)
wfximgfd.2 (𝜑𝐹:𝐴𝐵)
Assertion
Ref Expression
wfximgfd (𝜑 → (𝐹𝐶) ∈ (𝐹𝐴))

Proof of Theorem wfximgfd
StepHypRef Expression
1 wfximgfd.2 . . 3 (𝜑𝐹:𝐴𝐵)
21ffnd 6495 . 2 (𝜑𝐹 Fn 𝐴)
3 wfximgfd.1 . 2 (𝜑𝐶𝐴)
42, 3, 3fnfvimad 6979 1 (𝜑 → (𝐹𝐶) ∈ (𝐹𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2114   “ cima 5535  ⟶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-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-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:  imo72b2lem0  40802
 Copyright terms: Public domain W3C validator