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Theorem dfdm2 5908
Description: Alternate definition of domain df-dm 5352 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 5540 . . . . . 6 (𝐴𝐴) = (𝐴𝐴)
2 cocnvcnv2 5888 . . . . . 6 (𝐴𝐴) = (𝐴𝐴)
31, 2eqtri 2849 . . . . 5 (𝐴𝐴) = (𝐴𝐴)
43unieqi 4667 . . . 4 (𝐴𝐴) = (𝐴𝐴)
54unieqi 4667 . . 3 (𝐴𝐴) = (𝐴𝐴)
6 unidmrn 5906 . . 3 (𝐴𝐴) = (dom (𝐴𝐴) ∪ ran (𝐴𝐴))
75, 6eqtr3i 2851 . 2 (𝐴𝐴) = (dom (𝐴𝐴) ∪ ran (𝐴𝐴))
8 df-rn 5353 . . . . 5 ran 𝐴 = dom 𝐴
98eqcomi 2834 . . . 4 dom 𝐴 = ran 𝐴
10 dmcoeq 5621 . . . 4 (dom 𝐴 = ran 𝐴 → dom (𝐴𝐴) = dom 𝐴)
119, 10ax-mp 5 . . 3 dom (𝐴𝐴) = dom 𝐴
12 rncoeq 5622 . . . . 5 (dom 𝐴 = ran 𝐴 → ran (𝐴𝐴) = ran 𝐴)
139, 12ax-mp 5 . . . 4 ran (𝐴𝐴) = ran 𝐴
14 dfdm4 5548 . . . 4 dom 𝐴 = ran 𝐴
1513, 14eqtr4i 2852 . . 3 ran (𝐴𝐴) = dom 𝐴
1611, 15uneq12i 3992 . 2 (dom (𝐴𝐴) ∪ ran (𝐴𝐴)) = (dom 𝐴 ∪ dom 𝐴)
17 unidm 3983 . 2 (dom 𝐴 ∪ dom 𝐴) = dom 𝐴
187, 16, 173eqtrri 2854 1 dom 𝐴 = (𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1658  cun 3796   cuni 4658  ccnv 5341  dom cdm 5342  ran crn 5343  ccom 5346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354
This theorem is referenced by: (None)
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