Theorem imassrn 5093

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 1667	. . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴) → ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴)
2	1	ss2abi 3300	. 2 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)} ⊆ {𝑦 ∣ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴}
3		dfima3 5085	. 2 ⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}
4		dfrn3 4925	. 2 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴}
5	2, 3, 4	3sstr4i 3269	1 ⊢ (𝐴 “ 𝐵) ⊆ ran 𝐴

Syntax hints:   ∧ wa 104  ∃wex 1541   ∈ wcel 2202  {cab 2217   ⊆ wss 3201  ⟨cop 3676  ran crn 4732   “ cima 4734

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-cnv 4739  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744

This theorem is referenced by:  imaexg  5096  0ima  5103  cnvimass  5106  fimass  5505  fimacnv  5784  f1opw2  6239  smores2  6503  ecss  6788  f1imaen2g  7010  fopwdom  7065  ssenen  7080  phplem4dom  7091  isinfinf  7129  fiintim  7166  sbthlem2  7200  sbthlemi3  7201  sbthlemi5  7203  sbthlemi6  7204  ctssdccl  7353  ctinf  13112  ssnnctlemct  13128  mhmima  13635  cnptoprest2  15031  hmeontr  15104  hmeores  15106  tgqioo  15346  domomsubct  16703