MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnfvimad Structured version   Visualization version   GIF version

Theorem fnfvimad 7185
Description: A function's value belongs to the image. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
fnfvimad.1 (𝜑𝐹 Fn 𝐴)
fnfvimad.2 (𝜑𝐵𝐴)
fnfvimad.3 (𝜑𝐵𝐶)
Assertion
Ref Expression
fnfvimad (𝜑 → (𝐹𝐵) ∈ (𝐹𝐶))

Proof of Theorem fnfvimad
StepHypRef Expression
1 inss2 4190 . . 3 (𝐴𝐶) ⊆ 𝐶
2 imass2 6055 . . 3 ((𝐴𝐶) ⊆ 𝐶 → (𝐹 “ (𝐴𝐶)) ⊆ (𝐹𝐶))
31, 2ax-mp 5 . 2 (𝐹 “ (𝐴𝐶)) ⊆ (𝐹𝐶)
4 fnfvimad.1 . . 3 (𝜑𝐹 Fn 𝐴)
5 inss1 4189 . . . 4 (𝐴𝐶) ⊆ 𝐴
65a1i 11 . . 3 (𝜑 → (𝐴𝐶) ⊆ 𝐴)
7 fnfvimad.2 . . . 4 (𝜑𝐵𝐴)
8 fnfvimad.3 . . . 4 (𝜑𝐵𝐶)
97, 8elind 4155 . . 3 (𝜑𝐵 ∈ (𝐴𝐶))
10 fnfvima 7184 . . 3 ((𝐹 Fn 𝐴 ∧ (𝐴𝐶) ⊆ 𝐴𝐵 ∈ (𝐴𝐶)) → (𝐹𝐵) ∈ (𝐹 “ (𝐴𝐶)))
114, 6, 9, 10syl3anc 1372 . 2 (𝜑 → (𝐹𝐵) ∈ (𝐹 “ (𝐴𝐶)))
123, 11sselid 3943 1 (𝜑 → (𝐹𝐵) ∈ (𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  cin 3910  wss 3911  cima 5637   Fn wfn 6492  cfv 6497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-fv 6505
This theorem is referenced by:  cycpm3cl2  32034  rhmimaidl  32254  dimkerim  32379  wfximgfd  42524  limsupmnflem  44047  liminfval2  44095  limsup10exlem  44099  liminflelimsupuz  44112  fundcmpsurinjimaid  45689
  Copyright terms: Public domain W3C validator