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Theorem dfdm4 4554
Description: Alternate definition of domain. (Contributed by NM, 28-Dec-1996.)
Assertion
Ref Expression
dfdm4 dom 𝐴 = ran 𝐴

Proof of Theorem dfdm4
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2577 . . . . 5 𝑦 ∈ V
2 vex 2577 . . . . 5 𝑥 ∈ V
31, 2brcnv 4545 . . . 4 (𝑦𝐴𝑥𝑥𝐴𝑦)
43exbii 1512 . . 3 (∃𝑦 𝑦𝐴𝑥 ↔ ∃𝑦 𝑥𝐴𝑦)
54abbii 2169 . 2 {𝑥 ∣ ∃𝑦 𝑦𝐴𝑥} = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
6 dfrn2 4550 . 2 ran 𝐴 = {𝑥 ∣ ∃𝑦 𝑦𝐴𝑥}
7 df-dm 4382 . 2 dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
85, 6, 73eqtr4ri 2087 1 dom 𝐴 = ran 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1259  wex 1397  {cab 2042   class class class wbr 3791  ccnv 4371  dom cdm 4372  ran crn 4373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-cnv 4380  df-dm 4382  df-rn 4383
This theorem is referenced by:  dmcnvcnv  4585  rncnvcnv  4586  rncoeq  4632  cnvimass  4715  cnvimarndm  4716  dminxp  4792  cnvsn0  4816  rnsnopg  4826  dmmpt  4843  dmco  4856  cores2  4860  cnvssrndm  4869  unidmrn  4877  dfdm2  4879  cnvexg  4882  funimacnv  5002  foimacnv  5171  funcocnv2  5178  fimacnv  5323  f1opw2  5733  fopwdom  6340
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