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

Theorem fimass 5994
 Description: The image of a class is a subset of its codomain. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Assertion
Ref Expression
fimass (𝐹:𝐴𝐵 → (𝐹𝑋) ⊆ 𝐵)

Proof of Theorem fimass
StepHypRef Expression
1 imassrn 5396 . . 3 (𝐹𝑋) ⊆ ran 𝐹
21a1i 11 . 2 (𝐹:𝐴𝐵 → (𝐹𝑋) ⊆ ran 𝐹)
3 frn 5966 . 2 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
42, 3sstrd 3578 1 (𝐹:𝐴𝐵 → (𝐹𝑋) ⊆ 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ⊆ wss 3540  ran crn 5039   “ cima 5041  ⟶wf 5800 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-f 5808 This theorem is referenced by:  fissuni  8154  fipreima  8155  sge0f1o  39275  trlreslem  40907
 Copyright terms: Public domain W3C validator