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Theorem imadmrn 5932
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

Proof of Theorem imadmrn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3495 . . . . . . 7 𝑥 ∈ V
2 vex 3495 . . . . . . 7 𝑦 ∈ V
31, 2opeldm 5769 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 ∈ dom 𝐴)
43pm4.71i 560 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 ∈ dom 𝐴))
5 ancom 461 . . . . 5 ((⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 ∈ dom 𝐴) ↔ (𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
64, 5bitr2i 277 . . . 4 ((𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
76exbii 1839 . . 3 (∃𝑥(𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴) ↔ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴)
87abbii 2883 . 2 {𝑦 ∣ ∃𝑥(𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)} = {𝑦 ∣ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴}
9 dfima3 5925 . 2 (𝐴 “ dom 𝐴) = {𝑦 ∣ ∃𝑥(𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}
10 dfrn3 5753 . 2 ran 𝐴 = {𝑦 ∣ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴}
118, 9, 103eqtr4i 2851 1 (𝐴 “ dom 𝐴) = ran 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1528  wex 1771  wcel 2105  {cab 2796  cop 4563  dom cdm 5548  ran crn 5549  cima 5551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561
This theorem is referenced by:  cnvimarndm  5943  foima  6588  fimadmfo  6592  f1imacnv  6624  fsn2  6890  resfunexg  6969  elunirnALT  7002  fnexALT  7641  uniqs2  8348  mapsnd  8438  phplem4  8687  php3  8691  jech9.3  9231  fin4en1  9719  retopbas  23296  plyeq0  24728  rnelshi  29763  s2rn  30547  s3rn  30549  poimirlem3  34776  poimirlem30  34803
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