Theorem fimass 6738

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 6070	. 2 ⊢ (𝐹 “ 𝑋) ⊆ ran 𝐹
2		frn 6724	. 2 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
3	1, 2	sstrid 3993	1 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝑋) ⊆ 𝐵)

Syntax hints:   → wi 4   ⊆ wss 3948  ran crn 5677   “ cima 5679  ⟶wf 6539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-f 6547

This theorem is referenced by:  fimacnvOLD  7072  f1imaen2g  9013  domunsncan  9074  fissuni  9359  fipreima  9360  carduniima  10093  psgnunilem1  19402  fbasrn  23608  imaelfm  23675  wlkres  29182  trlreslem  29211  fimarab  32123  tocyccntz  32561  rhmimaidl  32812  nummin  34380  fimassd  44229  limsupvaluz  44723  sge0f1o  45397  fundcmpsurbijinjpreimafv  46374  fundcmpsurinjimaid  46378