Theorem fimass 6730

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 6197	. 2 ⊢ (𝐹 “ 𝑋) ⊆ ran 𝐹
2		frn 6717	. 2 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
3	1, 2	sstrid 3919	1 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝑋) ⊆ 𝐵)

This proof depends on syntax axioms:   → wi 4   ⊆ wss 3895  ran crn 5651   “ cima 5653  ⟶wf 6534

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-cnv 5658  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-f 6542

This theorem is used by:  fimassd  6731  fimarab  6959  f1imaen2g  9042  domunsncan  9096  fissuni  9346  fipreima  9347  carduniima  10175  psgnunilem1  19707  fbasrn  24203  imaelfm  24270  wlkres  30249  trlreslem  30282  tocyccntz  33705  rhmimaidl  33982  nummin  35722  dfscott3  35743  regsfromunir1  37328  hashscontpowcl  43171  relpfrlem  45943  fundcmpsurbijinjpreimafv  48489  fundcmpsurinjimaid  48493