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Theorem dfdm2 5636
Description: Alternate definition of domain df-dm 5094 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 5278 . . . . . 6 (𝐴𝐴) = (𝐴𝐴)
2 cocnvcnv2 5616 . . . . . 6 (𝐴𝐴) = (𝐴𝐴)
31, 2eqtri 2643 . . . . 5 (𝐴𝐴) = (𝐴𝐴)
43unieqi 4418 . . . 4 (𝐴𝐴) = (𝐴𝐴)
54unieqi 4418 . . 3 (𝐴𝐴) = (𝐴𝐴)
6 unidmrn 5634 . . 3 (𝐴𝐴) = (dom (𝐴𝐴) ∪ ran (𝐴𝐴))
75, 6eqtr3i 2645 . 2 (𝐴𝐴) = (dom (𝐴𝐴) ∪ ran (𝐴𝐴))
8 df-rn 5095 . . . . 5 ran 𝐴 = dom 𝐴
98eqcomi 2630 . . . 4 dom 𝐴 = ran 𝐴
10 dmcoeq 5358 . . . 4 (dom 𝐴 = ran 𝐴 → dom (𝐴𝐴) = dom 𝐴)
119, 10ax-mp 5 . . 3 dom (𝐴𝐴) = dom 𝐴
12 rncoeq 5359 . . . . 5 (dom 𝐴 = ran 𝐴 → ran (𝐴𝐴) = ran 𝐴)
139, 12ax-mp 5 . . . 4 ran (𝐴𝐴) = ran 𝐴
14 dfdm4 5286 . . . 4 dom 𝐴 = ran 𝐴
1513, 14eqtr4i 2646 . . 3 ran (𝐴𝐴) = dom 𝐴
1611, 15uneq12i 3749 . 2 (dom (𝐴𝐴) ∪ ran (𝐴𝐴)) = (dom 𝐴 ∪ dom 𝐴)
17 unidm 3740 . 2 (dom 𝐴 ∪ dom 𝐴) = dom 𝐴
187, 16, 173eqtrri 2648 1 dom 𝐴 = (𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  cun 3558   cuni 4409  ccnv 5083  dom cdm 5084  ran crn 5085  ccom 5088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096
This theorem is referenced by: (None)
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