Theorem dfdm2 6286

Description: Alternate definition of domain df-dm 5673 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.)

Assertion

Ref	Expression
dfdm2	⊢ dom 𝐴 = ∪ ∪ (◡𝐴 ∘ 𝐴)

Proof of Theorem dfdm2

Step	Hyp	Ref	Expression
1		cnvco 5877	. . . . . 6 ⊢ ◡(◡𝐴 ∘ 𝐴) = (◡𝐴 ∘ ◡◡𝐴)
2		cocnvcnv2 6262	. . . . . 6 ⊢ (◡𝐴 ∘ ◡◡𝐴) = (◡𝐴 ∘ 𝐴)
3	1, 2	eqtri 2788	. . . . 5 ⊢ ◡(◡𝐴 ∘ 𝐴) = (◡𝐴 ∘ 𝐴)
4	3	unieqi 4886	. . . 4 ⊢ ∪ ◡(◡𝐴 ∘ 𝐴) = ∪ (◡𝐴 ∘ 𝐴)
5	4	unieqi 4886	. . 3 ⊢ ∪ ∪ ◡(◡𝐴 ∘ 𝐴) = ∪ ∪ (◡𝐴 ∘ 𝐴)
6		unidmrn 6284	. . 3 ⊢ ∪ ∪ ◡(◡𝐴 ∘ 𝐴) = (dom (◡𝐴 ∘ 𝐴) ∪ ran (◡𝐴 ∘ 𝐴))
7	5, 6	eqtr3i 2790	. 2 ⊢ ∪ ∪ (◡𝐴 ∘ 𝐴) = (dom (◡𝐴 ∘ 𝐴) ∪ ran (◡𝐴 ∘ 𝐴))
8		df-rn 5674	. . . . 5 ⊢ ran 𝐴 = dom ◡𝐴
9	8	eqcomi 2774	. . . 4 ⊢ dom ◡𝐴 = ran 𝐴
10		dmcoeq 5972	. . . 4 ⊢ (dom ◡𝐴 = ran 𝐴 → dom (◡𝐴 ∘ 𝐴) = dom 𝐴)
11	9, 10	ax-mp 5	. . 3 ⊢ dom (◡𝐴 ∘ 𝐴) = dom 𝐴
12		rncoeq 5973	. . . . 5 ⊢ (dom ◡𝐴 = ran 𝐴 → ran (◡𝐴 ∘ 𝐴) = ran ◡𝐴)
13	9, 12	ax-mp 5	. . . 4 ⊢ ran (◡𝐴 ∘ 𝐴) = ran ◡𝐴
14		dfdm4 5887	. . . 4 ⊢ dom 𝐴 = ran ◡𝐴
15	13, 14	eqtr4i 2791	. . 3 ⊢ ran (◡𝐴 ∘ 𝐴) = dom 𝐴
16	11, 15	uneq12i 4120	. 2 ⊢ (dom (◡𝐴 ∘ 𝐴) ∪ ran (◡𝐴 ∘ 𝐴)) = (dom 𝐴 ∪ dom 𝐴)
17		unidm 4111	. 2 ⊢ (dom 𝐴 ∪ dom 𝐴) = dom 𝐴
18	7, 16, 17	3eqtrri 2793	1 ⊢ dom 𝐴 = ∪ ∪ (◡𝐴 ∘ 𝐴)

This proof depends on syntax axioms:   = wceq 1570   ∪ cun 3904  ∪ cuni 4874  ^◡ccnv 5662  dom cdm 5663  ran crn 5664   ∘ ccom 5667

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675

This theorem is used by: (None)