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Theorem dfdm2 5705
Description: Alternate definition of domain df-dm 5153 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 5340 . . . . . 6 (𝐴𝐴) = (𝐴𝐴)
2 cocnvcnv2 5685 . . . . . 6 (𝐴𝐴) = (𝐴𝐴)
31, 2eqtri 2673 . . . . 5 (𝐴𝐴) = (𝐴𝐴)
43unieqi 4477 . . . 4 (𝐴𝐴) = (𝐴𝐴)
54unieqi 4477 . . 3 (𝐴𝐴) = (𝐴𝐴)
6 unidmrn 5703 . . 3 (𝐴𝐴) = (dom (𝐴𝐴) ∪ ran (𝐴𝐴))
75, 6eqtr3i 2675 . 2 (𝐴𝐴) = (dom (𝐴𝐴) ∪ ran (𝐴𝐴))
8 df-rn 5154 . . . . 5 ran 𝐴 = dom 𝐴
98eqcomi 2660 . . . 4 dom 𝐴 = ran 𝐴
10 dmcoeq 5420 . . . 4 (dom 𝐴 = ran 𝐴 → dom (𝐴𝐴) = dom 𝐴)
119, 10ax-mp 5 . . 3 dom (𝐴𝐴) = dom 𝐴
12 rncoeq 5421 . . . . 5 (dom 𝐴 = ran 𝐴 → ran (𝐴𝐴) = ran 𝐴)
139, 12ax-mp 5 . . . 4 ran (𝐴𝐴) = ran 𝐴
14 dfdm4 5348 . . . 4 dom 𝐴 = ran 𝐴
1513, 14eqtr4i 2676 . . 3 ran (𝐴𝐴) = dom 𝐴
1611, 15uneq12i 3798 . 2 (dom (𝐴𝐴) ∪ ran (𝐴𝐴)) = (dom 𝐴 ∪ dom 𝐴)
17 unidm 3789 . 2 (dom 𝐴 ∪ dom 𝐴) = dom 𝐴
187, 16, 173eqtrri 2678 1 dom 𝐴 = (𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  cun 3605   cuni 4468  ccnv 5142  dom cdm 5143  ran crn 5144  ccom 5147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155
This theorem is referenced by: (None)
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