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Theorem imassrn 4679
 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 1509 . . 3 (∃𝑥(𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴) → ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴)
21ss2abi 3012 . 2 {𝑦 ∣ ∃𝑥(𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)} ⊆ {𝑦 ∣ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴}
3 dfima3 4671 . 2 (𝐴𝐵) = {𝑦 ∣ ∃𝑥(𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}
4 dfrn3 4524 . 2 ran 𝐴 = {𝑦 ∣ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴}
52, 3, 43sstr4i 2984 1 (𝐴𝐵) ⊆ ran 𝐴
 Colors of variables: wff set class Syntax hints:   ∧ wa 97  ∃wex 1381   ∈ wcel 1393  {cab 2026   ⊆ wss 2917  ⟨cop 3378  ran crn 4346   “ cima 4348 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358 This theorem is referenced by:  imaexg  4680  0ima  4685  cnvimass  4688  fimacnv  5296  f1opw2  5706  smores2  5909  ecss  6147  f1imaen2g  6273  fopwdom  6310  phplem4dom  6324
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