Theorem imassrn 5085

Description: The image of a class is a subset of its range. Theorem 3.16(xi) of [Monk1] p. 39. (Contributed by NM, 31-Mar-1995.)

Assertion

Ref	Expression
imassrn	⊢ (𝐴 “ 𝐵) ⊆ ran 𝐴

Proof of Theorem imassrn

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		exsimpr 1664	. . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴) → ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴)
2	1	ss2abi 3297	. 2 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)} ⊆ {𝑦 ∣ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴}
3		dfima3 5077	. 2 ⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}
4		dfrn3 4917	. 2 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴}
5	2, 3, 4	3sstr4i 3266	1 ⊢ (𝐴 “ 𝐵) ⊆ ran 𝐴

Syntax hints:   ∧ wa 104  ∃wex 1538   ∈ wcel 2200  {cab 2215   ⊆ wss 3198  ⟨cop 3670  ran crn 4724   “ cima 4726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-xp 4729  df-cnv 4731  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736

This theorem is referenced by:  imaexg  5088  0ima  5094  cnvimass  5097  fimass  5495  fimacnv  5772  f1opw2  6224  smores2  6455  ecss  6740  f1imaen2g  6962  fopwdom  7017  ssenen  7032  phplem4dom  7043  isinfinf  7081  fiintim  7118  sbthlem2  7151  sbthlemi3  7152  sbthlemi5  7154  sbthlemi6  7155  ctssdccl  7304  ctinf  13044  ssnnctlemct  13060  mhmima  13567  cnptoprest2  14957  hmeontr  15030  hmeores  15032  tgqioo  15272  domomsubct  16552