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Theorem dfdm2 6119
Description: Alternate definition of domain df-dm 5552 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 5743 . . . . . 6 (𝐴𝐴) = (𝐴𝐴)
2 cocnvcnv2 6098 . . . . . 6 (𝐴𝐴) = (𝐴𝐴)
31, 2eqtri 2847 . . . . 5 (𝐴𝐴) = (𝐴𝐴)
43unieqi 4837 . . . 4 (𝐴𝐴) = (𝐴𝐴)
54unieqi 4837 . . 3 (𝐴𝐴) = (𝐴𝐴)
6 unidmrn 6117 . . 3 (𝐴𝐴) = (dom (𝐴𝐴) ∪ ran (𝐴𝐴))
75, 6eqtr3i 2849 . 2 (𝐴𝐴) = (dom (𝐴𝐴) ∪ ran (𝐴𝐴))
8 df-rn 5553 . . . . 5 ran 𝐴 = dom 𝐴
98eqcomi 2833 . . . 4 dom 𝐴 = ran 𝐴
10 dmcoeq 5832 . . . 4 (dom 𝐴 = ran 𝐴 → dom (𝐴𝐴) = dom 𝐴)
119, 10ax-mp 5 . . 3 dom (𝐴𝐴) = dom 𝐴
12 rncoeq 5833 . . . . 5 (dom 𝐴 = ran 𝐴 → ran (𝐴𝐴) = ran 𝐴)
139, 12ax-mp 5 . . . 4 ran (𝐴𝐴) = ran 𝐴
14 dfdm4 5751 . . . 4 dom 𝐴 = ran 𝐴
1513, 14eqtr4i 2850 . . 3 ran (𝐴𝐴) = dom 𝐴
1611, 15uneq12i 4123 . 2 (dom (𝐴𝐴) ∪ ran (𝐴𝐴)) = (dom 𝐴 ∪ dom 𝐴)
17 unidm 4114 . 2 (dom 𝐴 ∪ dom 𝐴) = dom 𝐴
187, 16, 173eqtrri 2852 1 dom 𝐴 = (𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  cun 3917   cuni 4824  ccnv 5541  dom cdm 5542  ran crn 5543  ccom 5546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554
This theorem is referenced by: (None)
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