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Theorem imadmrn 4605
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 2538 . . . . . . 7 x V
2 vex 2538 . . . . . . 7 y V
31, 2opeldm 4465 . . . . . 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 1478 . . 3 (x(x dom A x, y A) ↔ xx, y A)
87abbii 2135 . 2 {yx(x dom A x, y A)} = {yxx, y A}
9 dfima3 4598 . 2 (A “ dom A) = {yx(x dom A x, y A)}
10 dfrn3 4451 . 2 ran A = {yxx, y A}
118, 9, 103eqtr4i 2052 1 (A “ dom A) = ran A
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1228  wex 1362   wcel 1374  {cab 2008  cop 3353  dom cdm 4272  ran crn 4273  cima 4275
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-xp 4278  df-cnv 4280  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285
This theorem is referenced by:  cnvimarndm  4616  foima  5036  f1imacnv  5068  fsn2  5262  resfunexg  5307  funiunfvdm  5327  fnexALT  5663  uniqs2  6077
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