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Theorem imadmrn 5395
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 3176 . . . . . . 7 𝑥 ∈ V
2 vex 3176 . . . . . . 7 𝑦 ∈ V
31, 2opeldm 5250 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 ∈ dom 𝐴)
43pm4.71i 662 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 ∈ dom 𝐴))
5 ancom 465 . . . . 5 ((⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 ∈ dom 𝐴) ↔ (𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
64, 5bitr2i 264 . . . 4 ((𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
76exbii 1764 . . 3 (∃𝑥(𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴) ↔ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴)
87abbii 2726 . 2 {𝑦 ∣ ∃𝑥(𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)} = {𝑦 ∣ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴}
9 dfima3 5388 . 2 (𝐴 “ dom 𝐴) = {𝑦 ∣ ∃𝑥(𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}
10 dfrn3 5234 . 2 ran 𝐴 = {𝑦 ∣ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴}
118, 9, 103eqtr4i 2642 1 (𝐴 “ dom 𝐴) = ran 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1475  wex 1695  wcel 1977  {cab 2596  cop 4131  dom cdm 5038  ran crn 5039  cima 5041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051
This theorem is referenced by:  cnvimarndm  5405  foima  6033  f1imacnv  6066  fsn2  6309  resfunexg  6384  elunirnALT  6414  fnexALT  7025  uniqs2  7696  mapsn  7785  phplem4  8027  php3  8031  jech9.3  8560  fin4en1  9014  retopbas  22374  plyeq0  23771  rnelshi  28302  poimirlem3  32582  poimirlem30  32609  mapsnd  38383
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