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Theorem dfdm2 6234
Description: Alternate definition of domain df-dm 5644 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 5842 . . . . . 6 (𝐴𝐴) = (𝐴𝐴)
2 cocnvcnv2 6211 . . . . . 6 (𝐴𝐴) = (𝐴𝐴)
31, 2eqtri 2761 . . . . 5 (𝐴𝐴) = (𝐴𝐴)
43unieqi 4879 . . . 4 (𝐴𝐴) = (𝐴𝐴)
54unieqi 4879 . . 3 (𝐴𝐴) = (𝐴𝐴)
6 unidmrn 6232 . . 3 (𝐴𝐴) = (dom (𝐴𝐴) ∪ ran (𝐴𝐴))
75, 6eqtr3i 2763 . 2 (𝐴𝐴) = (dom (𝐴𝐴) ∪ ran (𝐴𝐴))
8 df-rn 5645 . . . . 5 ran 𝐴 = dom 𝐴
98eqcomi 2742 . . . 4 dom 𝐴 = ran 𝐴
10 dmcoeq 5930 . . . 4 (dom 𝐴 = ran 𝐴 → dom (𝐴𝐴) = dom 𝐴)
119, 10ax-mp 5 . . 3 dom (𝐴𝐴) = dom 𝐴
12 rncoeq 5931 . . . . 5 (dom 𝐴 = ran 𝐴 → ran (𝐴𝐴) = ran 𝐴)
139, 12ax-mp 5 . . . 4 ran (𝐴𝐴) = ran 𝐴
14 dfdm4 5852 . . . 4 dom 𝐴 = ran 𝐴
1513, 14eqtr4i 2764 . . 3 ran (𝐴𝐴) = dom 𝐴
1611, 15uneq12i 4122 . 2 (dom (𝐴𝐴) ∪ ran (𝐴𝐴)) = (dom 𝐴 ∪ dom 𝐴)
17 unidm 4113 . 2 (dom 𝐴 ∪ dom 𝐴) = dom 𝐴
187, 16, 173eqtrri 2766 1 dom 𝐴 = (𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  cun 3909   cuni 4866  ccnv 5633  dom cdm 5634  ran crn 5635  ccom 5638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646
This theorem is referenced by: (None)
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