Theorem fimass 6690

Description: The image of a class under a function with domain and codomain is a subset of its codomain. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Assertion

Ref	Expression
fimass	⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝑋) ⊆ 𝐵)

Proof of Theorem fimass

Step	Hyp	Ref	Expression
1		imassrn 6031	. 2 ⊢ (𝐹 “ 𝑋) ⊆ ran 𝐹
2		frn 6677	. 2 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
3	1, 2	sstrid 3955	1 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝑋) ⊆ 𝐵)

Syntax hints:   → wi 4   ⊆ wss 3911  ran crn 5632   “ cima 5634  ⟶wf 6495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-f 6503

This theorem is referenced by:  fimassd  6691  fimarab  6917  f1imaen2g  8963  domunsncan  9018  fissuni  9284  fipreima  9285  carduniima  10025  psgnunilem1  19407  fbasrn  23804  imaelfm  23871  wlkres  29649  trlreslem  29678  tocyccntz  33116  rhmimaidl  33396  nummin  35074  hashscontpowcl  42101  relpfrlem  44936  limsupvaluz  45699  fundcmpsurbijinjpreimafv  47401  fundcmpsurinjimaid  47405