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 Description: The image of the domain of a class is the range of the class. (Contributed by NM, 14-Aug-1994.)
Assertion
Ref Expression
imadmrn (𝐴 “ dom 𝐴) = ran 𝐴

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2560 . . . . . . 7 𝑥 ∈ V
2 vex 2560 . . . . . . 7 𝑦 ∈ V
31, 2opeldm 4538 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 ∈ dom 𝐴)
43pm4.71i 371 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 ∈ dom 𝐴))
5 ancom 253 . . . . 5 ((⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 ∈ dom 𝐴) ↔ (𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
64, 5bitr2i 174 . . . 4 ((𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
76exbii 1496 . . 3 (∃𝑥(𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴) ↔ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴)
87abbii 2153 . 2 {𝑦 ∣ ∃𝑥(𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)} = {𝑦 ∣ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴}
9 dfima3 4671 . 2 (𝐴 “ dom 𝐴) = {𝑦 ∣ ∃𝑥(𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}
10 dfrn3 4524 . 2 ran 𝐴 = {𝑦 ∣ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴}
118, 9, 103eqtr4i 2070 1 (𝐴 “ dom 𝐴) = ran 𝐴
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1243  ∃wex 1381   ∈ wcel 1393  {cab 2026  ⟨cop 3378  dom cdm 4345  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:  cnvimarndm  4689  foima  5111  f1imacnv  5143  fsn2  5337  resfunexg  5382  funiunfvdm  5402  fnexALT  5740  uniqs2  6166  phplem4  6318  phplem4on  6329
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