Theorem fimass 6727

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 6074	. 2 ⊢ (𝐹 “ 𝑋) ⊆ ran 𝐹
2		frn 6714	. 2 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
3	1, 2	sstrid 3956	1 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝑋) ⊆ 𝐵)

Syntax hints:   → wi 4   ⊆ wss 3913  ran crn 5663   “ cima 5665  ⟶wf 6533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-xp 5668  df-cnv 5670  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-f 6541

This theorem is referenced by:  fimassd  6728  fimarab  6956  f1imaen2g  9012  domunsncan  9065  fissuni  9314  fipreima  9315  carduniima  10080  psgnunilem1  19563  fbasrn  24010  imaelfm  24077  wlkres  29959  trlreslem  29988  tocyccntz  33405  rhmimaidl  33684  nummin  35427  regsfromunir1  36974  hashscontpowcl  42811  relpfrlem  45588  fundcmpsurbijinjpreimafv  48079  fundcmpsurinjimaid  48083