Theorem fimass 6726

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 6058	. 2 ⊢ (𝐹 “ 𝑋) ⊆ ran 𝐹
2		frn 6713	. 2 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
3	1, 2	sstrid 3970	1 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝑋) ⊆ 𝐵)

Syntax hints:   → wi 4   ⊆ wss 3926  ran crn 5655   “ cima 5657  ⟶wf 6527

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

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 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-xp 5660  df-cnv 5662  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-f 6535

This theorem is referenced by:  fimassd  6727  fimarab  6953  f1imaen2g  9029  domunsncan  9086  fissuni  9369  fipreima  9370  carduniima  10110  psgnunilem1  19474  fbasrn  23822  imaelfm  23889  wlkres  29650  trlreslem  29679  tocyccntz  33155  rhmimaidl  33447  nummin  35122  hashscontpowcl  42133  relpfrlem  44978  limsupvaluz  45737  fundcmpsurbijinjpreimafv  47421  fundcmpsurinjimaid  47425