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Theorem dfdm2 6303
Description: Alternate definition of domain df-dm 5699 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 5899 . . . . . 6 (𝐴𝐴) = (𝐴𝐴)
2 cocnvcnv2 6280 . . . . . 6 (𝐴𝐴) = (𝐴𝐴)
31, 2eqtri 2763 . . . . 5 (𝐴𝐴) = (𝐴𝐴)
43unieqi 4924 . . . 4 (𝐴𝐴) = (𝐴𝐴)
54unieqi 4924 . . 3 (𝐴𝐴) = (𝐴𝐴)
6 unidmrn 6301 . . 3 (𝐴𝐴) = (dom (𝐴𝐴) ∪ ran (𝐴𝐴))
75, 6eqtr3i 2765 . 2 (𝐴𝐴) = (dom (𝐴𝐴) ∪ ran (𝐴𝐴))
8 df-rn 5700 . . . . 5 ran 𝐴 = dom 𝐴
98eqcomi 2744 . . . 4 dom 𝐴 = ran 𝐴
10 dmcoeq 5992 . . . 4 (dom 𝐴 = ran 𝐴 → dom (𝐴𝐴) = dom 𝐴)
119, 10ax-mp 5 . . 3 dom (𝐴𝐴) = dom 𝐴
12 rncoeq 5993 . . . . 5 (dom 𝐴 = ran 𝐴 → ran (𝐴𝐴) = ran 𝐴)
139, 12ax-mp 5 . . . 4 ran (𝐴𝐴) = ran 𝐴
14 dfdm4 5909 . . . 4 dom 𝐴 = ran 𝐴
1513, 14eqtr4i 2766 . . 3 ran (𝐴𝐴) = dom 𝐴
1611, 15uneq12i 4176 . 2 (dom (𝐴𝐴) ∪ ran (𝐴𝐴)) = (dom 𝐴 ∪ dom 𝐴)
17 unidm 4167 . 2 (dom 𝐴 ∪ dom 𝐴) = dom 𝐴
187, 16, 173eqtrri 2768 1 dom 𝐴 = (𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  cun 3961   cuni 4912  ccnv 5688  dom cdm 5689  ran crn 5690  ccom 5693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701
This theorem is referenced by: (None)
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