Theorem dfdm2 6239

Description: Alternate definition of domain df-dm 5634 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 5834	. . . . . 6 ⊢ ◡(◡𝐴 ∘ 𝐴) = (◡𝐴 ∘ ◡◡𝐴)
2		cocnvcnv2 6217	. . . . . 6 ⊢ (◡𝐴 ∘ ◡◡𝐴) = (◡𝐴 ∘ 𝐴)
3	1, 2	eqtri 2759	. . . . 5 ⊢ ◡(◡𝐴 ∘ 𝐴) = (◡𝐴 ∘ 𝐴)
4	3	unieqi 4875	. . . 4 ⊢ ∪ ◡(◡𝐴 ∘ 𝐴) = ∪ (◡𝐴 ∘ 𝐴)
5	4	unieqi 4875	. . 3 ⊢ ∪ ∪ ◡(◡𝐴 ∘ 𝐴) = ∪ ∪ (◡𝐴 ∘ 𝐴)
6		unidmrn 6237	. . 3 ⊢ ∪ ∪ ◡(◡𝐴 ∘ 𝐴) = (dom (◡𝐴 ∘ 𝐴) ∪ ran (◡𝐴 ∘ 𝐴))
7	5, 6	eqtr3i 2761	. 2 ⊢ ∪ ∪ (◡𝐴 ∘ 𝐴) = (dom (◡𝐴 ∘ 𝐴) ∪ ran (◡𝐴 ∘ 𝐴))
8		df-rn 5635	. . . . 5 ⊢ ran 𝐴 = dom ◡𝐴
9	8	eqcomi 2745	. . . 4 ⊢ dom ◡𝐴 = ran 𝐴
10		dmcoeq 5930	. . . 4 ⊢ (dom ◡𝐴 = ran 𝐴 → dom (◡𝐴 ∘ 𝐴) = dom 𝐴)
11	9, 10	ax-mp 5	. . 3 ⊢ dom (◡𝐴 ∘ 𝐴) = dom 𝐴
12		rncoeq 5931	. . . . 5 ⊢ (dom ◡𝐴 = ran 𝐴 → ran (◡𝐴 ∘ 𝐴) = ran ◡𝐴)
13	9, 12	ax-mp 5	. . . 4 ⊢ ran (◡𝐴 ∘ 𝐴) = ran ◡𝐴
14		dfdm4 5844	. . . 4 ⊢ dom 𝐴 = ran ◡𝐴
15	13, 14	eqtr4i 2762	. . 3 ⊢ ran (◡𝐴 ∘ 𝐴) = dom 𝐴
16	11, 15	uneq12i 4118	. 2 ⊢ (dom (◡𝐴 ∘ 𝐴) ∪ ran (◡𝐴 ∘ 𝐴)) = (dom 𝐴 ∪ dom 𝐴)
17		unidm 4109	. 2 ⊢ (dom 𝐴 ∪ dom 𝐴) = dom 𝐴
18	7, 16, 17	3eqtrri 2764	1 ⊢ dom 𝐴 = ∪ ∪ (◡𝐴 ∘ 𝐴)

Syntax hints:   = wceq 1541   ∪ cun 3899  ∪ cuni 4863  ^◡ccnv 5623  dom cdm 5624  ran crn 5625   ∘ ccom 5628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636

This theorem is referenced by: (None)