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 6069	. 2 ⊢ (𝐹 “ 𝑋) ⊆ ran 𝐹
2		frn 6713	. 2 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
3	1, 2	sstrid 3942	1 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝑋) ⊆ 𝐵)

This proof depends on syntax axioms:   → wi 4   ⊆ wss 3899  ran crn 5656   “ cima 5658  ⟶wf 6531

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 2732  ax-sep 5251  ax-pr 5398

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 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5661  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-f 6539

This theorem is used by:  fimassd  6727  fimarab  6955  f1imaen2g  9028  domunsncan  9082  fissuni  9331  fipreima  9332  carduniima  10124  psgnunilem1  19646  fbasrn  24142  imaelfm  24209  wlkres  30160  trlreslem  30193  tocyccntz  33616  rhmimaidl  33893  nummin  35631  dfscott3  35659  regsfromunir1  37226  hashscontpowcl  43051  relpfrlem  45841  fundcmpsurbijinjpreimafv  48372  fundcmpsurinjimaid  48376