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Theorem imadmrn 4621
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 2554 . . . . . . 7 x V
2 vex 2554 . . . . . . 7 y V
31, 2opeldm 4481 . . . . . 6 (⟨x, y Ax dom A)
43pm4.71i 371 . . . . 5 (⟨x, y A ↔ (⟨x, y A x dom A))
5 ancom 253 . . . . 5 ((⟨x, y A x dom A) ↔ (x dom A x, y A))
64, 5bitr2i 174 . . . 4 ((x dom A x, y A) ↔ ⟨x, y A)
76exbii 1493 . . 3 (x(x dom A x, y A) ↔ xx, y A)
87abbii 2150 . 2 {yx(x dom A x, y A)} = {yxx, y A}
9 dfima3 4614 . 2 (A “ dom A) = {yx(x dom A x, y A)}
10 dfrn3 4467 . 2 ran A = {yxx, y A}
118, 9, 103eqtr4i 2067 1 (A “ dom A) = ran A
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242  wex 1378   wcel 1390  {cab 2023  cop 3370  dom cdm 4288  ran crn 4289  cima 4291
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301
This theorem is referenced by:  cnvimarndm  4632  foima  5054  f1imacnv  5086  fsn2  5280  resfunexg  5325  funiunfvdm  5345  fnexALT  5682  uniqs2  6102
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