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 6071	. 2 ⊢ (𝐹 “ 𝑋) ⊆ ran 𝐹
2		frn 6714	. 2 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
3	1, 2	sstrid 3945	1 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝑋) ⊆ 𝐵)

This proof depends on syntax axioms:   → wi 4   ⊆ wss 3902  ran crn 5660   “ cima 5662  ⟶wf 6533

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-f 6541

This theorem is used by:  fimassd  6728  fimarab  6956  f1imaen2g  9024  domunsncan  9078  fissuni  9327  fipreima  9328  carduniima  10102  psgnunilem1  19619  fbasrn  24109  imaelfm  24176  wlkres  30112  trlreslem  30145  tocyccntz  33569  rhmimaidl  33845  nummin  35583  dfscott3  35611  regsfromunir1  37144  hashscontpowcl  42971  relpfrlem  45761  fundcmpsurbijinjpreimafv  48292  fundcmpsurinjimaid  48296