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Theorem dfdm2 5921
 Description: Alternate definition of domain df-dm 5365 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 5553 . . . . . 6 (𝐴𝐴) = (𝐴𝐴)
2 cocnvcnv2 5901 . . . . . 6 (𝐴𝐴) = (𝐴𝐴)
31, 2eqtri 2801 . . . . 5 (𝐴𝐴) = (𝐴𝐴)
43unieqi 4680 . . . 4 (𝐴𝐴) = (𝐴𝐴)
54unieqi 4680 . . 3 (𝐴𝐴) = (𝐴𝐴)
6 unidmrn 5919 . . 3 (𝐴𝐴) = (dom (𝐴𝐴) ∪ ran (𝐴𝐴))
75, 6eqtr3i 2803 . 2 (𝐴𝐴) = (dom (𝐴𝐴) ∪ ran (𝐴𝐴))
8 df-rn 5366 . . . . 5 ran 𝐴 = dom 𝐴
98eqcomi 2786 . . . 4 dom 𝐴 = ran 𝐴
10 dmcoeq 5634 . . . 4 (dom 𝐴 = ran 𝐴 → dom (𝐴𝐴) = dom 𝐴)
119, 10ax-mp 5 . . 3 dom (𝐴𝐴) = dom 𝐴
12 rncoeq 5635 . . . . 5 (dom 𝐴 = ran 𝐴 → ran (𝐴𝐴) = ran 𝐴)
139, 12ax-mp 5 . . . 4 ran (𝐴𝐴) = ran 𝐴
14 dfdm4 5561 . . . 4 dom 𝐴 = ran 𝐴
1513, 14eqtr4i 2804 . . 3 ran (𝐴𝐴) = dom 𝐴
1611, 15uneq12i 3987 . 2 (dom (𝐴𝐴) ∪ ran (𝐴𝐴)) = (dom 𝐴 ∪ dom 𝐴)
17 unidm 3978 . 2 (dom 𝐴 ∪ dom 𝐴) = dom 𝐴
187, 16, 173eqtrri 2806 1 dom 𝐴 = (𝐴𝐴)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1601   ∪ cun 3789  ∪ cuni 4671  ◡ccnv 5354  dom cdm 5355  ran crn 5356   ∘ ccom 5359 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367 This theorem is referenced by: (None)
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