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Theorem imadmrn 4975
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 2740 . . . . . . 7 𝑥 ∈ V
2 vex 2740 . . . . . . 7 𝑦 ∈ V
31, 2opeldm 4825 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 ∈ dom 𝐴)
43pm4.71i 391 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 ∈ dom 𝐴))
5 ancom 266 . . . . 5 ((⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 ∈ dom 𝐴) ↔ (𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
64, 5bitr2i 185 . . . 4 ((𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
76exbii 1605 . . 3 (∃𝑥(𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴) ↔ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴)
87abbii 2293 . 2 {𝑦 ∣ ∃𝑥(𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)} = {𝑦 ∣ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴}
9 dfima3 4968 . 2 (𝐴 “ dom 𝐴) = {𝑦 ∣ ∃𝑥(𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}
10 dfrn3 4811 . 2 ran 𝐴 = {𝑦 ∣ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴}
118, 9, 103eqtr4i 2208 1 (𝐴 “ dom 𝐴) = ran 𝐴
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1353  wex 1492  wcel 2148  {cab 2163  cop 3594  dom cdm 4622  ran crn 4623  cima 4625
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-xp 4628  df-cnv 4630  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635
This theorem is referenced by:  cnvimarndm  4987  foima  5438  f1imacnv  5473  fsn2  5685  resfunexg  5732  funiunfvdm  5757  fnexALT  6105  uniqs2  6588  mapsn  6683  phplem4  6848  phplem4on  6860  retopbas  13656
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