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Theorem dfdm2 6125
Description: Alternate definition of domain df-dm 5558 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 5749 . . . . . 6 (𝐴𝐴) = (𝐴𝐴)
2 cocnvcnv2 6104 . . . . . 6 (𝐴𝐴) = (𝐴𝐴)
31, 2eqtri 2841 . . . . 5 (𝐴𝐴) = (𝐴𝐴)
43unieqi 4839 . . . 4 (𝐴𝐴) = (𝐴𝐴)
54unieqi 4839 . . 3 (𝐴𝐴) = (𝐴𝐴)
6 unidmrn 6123 . . 3 (𝐴𝐴) = (dom (𝐴𝐴) ∪ ran (𝐴𝐴))
75, 6eqtr3i 2843 . 2 (𝐴𝐴) = (dom (𝐴𝐴) ∪ ran (𝐴𝐴))
8 df-rn 5559 . . . . 5 ran 𝐴 = dom 𝐴
98eqcomi 2827 . . . 4 dom 𝐴 = ran 𝐴
10 dmcoeq 5838 . . . 4 (dom 𝐴 = ran 𝐴 → dom (𝐴𝐴) = dom 𝐴)
119, 10ax-mp 5 . . 3 dom (𝐴𝐴) = dom 𝐴
12 rncoeq 5839 . . . . 5 (dom 𝐴 = ran 𝐴 → ran (𝐴𝐴) = ran 𝐴)
139, 12ax-mp 5 . . . 4 ran (𝐴𝐴) = ran 𝐴
14 dfdm4 5757 . . . 4 dom 𝐴 = ran 𝐴
1513, 14eqtr4i 2844 . . 3 ran (𝐴𝐴) = dom 𝐴
1611, 15uneq12i 4134 . 2 (dom (𝐴𝐴) ∪ ran (𝐴𝐴)) = (dom 𝐴 ∪ dom 𝐴)
17 unidm 4125 . 2 (dom 𝐴 ∪ dom 𝐴) = dom 𝐴
187, 16, 173eqtrri 2846 1 dom 𝐴 = (𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  cun 3931   cuni 4830  ccnv 5547  dom cdm 5548  ran crn 5549  ccom 5552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560
This theorem is referenced by: (None)
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