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Theorem dfdm2 5029
Description: Alternate definition of domain df-dm 4507 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.)
Assertion
Ref Expression
dfdm2 dom 𝐴 = (𝐴𝐴)

Proof of Theorem dfdm2
StepHypRef Expression
1 cnvco 4682 . . . . . 6 (𝐴𝐴) = (𝐴𝐴)
2 cocnvcnv2 5006 . . . . . 6 (𝐴𝐴) = (𝐴𝐴)
31, 2eqtri 2133 . . . . 5 (𝐴𝐴) = (𝐴𝐴)
43unieqi 3710 . . . 4 (𝐴𝐴) = (𝐴𝐴)
54unieqi 3710 . . 3 (𝐴𝐴) = (𝐴𝐴)
6 unidmrn 5027 . . 3 (𝐴𝐴) = (dom (𝐴𝐴) ∪ ran (𝐴𝐴))
75, 6eqtr3i 2135 . 2 (𝐴𝐴) = (dom (𝐴𝐴) ∪ ran (𝐴𝐴))
8 df-rn 4508 . . . . 5 ran 𝐴 = dom 𝐴
98eqcomi 2117 . . . 4 dom 𝐴 = ran 𝐴
10 dmcoeq 4767 . . . 4 (dom 𝐴 = ran 𝐴 → dom (𝐴𝐴) = dom 𝐴)
119, 10ax-mp 7 . . 3 dom (𝐴𝐴) = dom 𝐴
12 rncoeq 4768 . . . . 5 (dom 𝐴 = ran 𝐴 → ran (𝐴𝐴) = ran 𝐴)
139, 12ax-mp 7 . . . 4 ran (𝐴𝐴) = ran 𝐴
14 dfdm4 4689 . . . 4 dom 𝐴 = ran 𝐴
1513, 14eqtr4i 2136 . . 3 ran (𝐴𝐴) = dom 𝐴
1611, 15uneq12i 3192 . 2 (dom (𝐴𝐴) ∪ ran (𝐴𝐴)) = (dom 𝐴 ∪ dom 𝐴)
17 unidm 3183 . 2 (dom 𝐴 ∪ dom 𝐴) = dom 𝐴
187, 16, 173eqtrri 2138 1 dom 𝐴 = (𝐴𝐴)
Colors of variables: wff set class
Syntax hints:   = wceq 1312  cun 3033   cuni 3700  ccnv 4496  dom cdm 4497  ran crn 4498  ccom 4501
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509
This theorem is referenced by: (None)
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