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Theorem imassrn 5016
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 (𝐴𝐵) ⊆ ran 𝐴

Proof of Theorem imassrn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 exsimpr 1629 . . 3 (∃𝑥(𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴) → ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴)
21ss2abi 3251 . 2 {𝑦 ∣ ∃𝑥(𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)} ⊆ {𝑦 ∣ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴}
3 dfima3 5008 . 2 (𝐴𝐵) = {𝑦 ∣ ∃𝑥(𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}
4 dfrn3 4851 . 2 ran 𝐴 = {𝑦 ∣ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴}
52, 3, 43sstr4i 3220 1 (𝐴𝐵) ⊆ ran 𝐴
Colors of variables: wff set class
Syntax hints:  wa 104  wex 1503  wcel 2164  {cab 2179  wss 3153  cop 3621  ran crn 4660  cima 4662
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-cnv 4667  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672
This theorem is referenced by:  imaexg  5019  0ima  5025  cnvimass  5028  fimacnv  5687  f1opw2  6124  smores2  6347  ecss  6630  f1imaen2g  6847  fopwdom  6892  ssenen  6907  phplem4dom  6918  isinfinf  6953  fiintim  6985  sbthlem2  7017  sbthlemi3  7018  sbthlemi5  7020  sbthlemi6  7021  ctssdccl  7170  ctinf  12587  ssnnctlemct  12603  mhmima  13063  cnptoprest2  14408  hmeontr  14481  hmeores  14483  tgqioo  14715
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