Theorem dfdm2 6268

Description: Alternate definition of domain df-dm 5657 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 5861	. . . . . 6 ⊢ ◡(◡𝐴 ∘ 𝐴) = (◡𝐴 ∘ ◡◡𝐴)
2		cocnvcnv2 6246	. . . . . 6 ⊢ (◡𝐴 ∘ ◡◡𝐴) = (◡𝐴 ∘ 𝐴)
3	1, 2	eqtri 2785	. . . . 5 ⊢ ◡(◡𝐴 ∘ 𝐴) = (◡𝐴 ∘ 𝐴)
4	3	unieqi 4877	. . . 4 ⊢ ∪ ◡(◡𝐴 ∘ 𝐴) = ∪ (◡𝐴 ∘ 𝐴)
5	4	unieqi 4877	. . 3 ⊢ ∪ ∪ ◡(◡𝐴 ∘ 𝐴) = ∪ ∪ (◡𝐴 ∘ 𝐴)
6		unidmrn 6266	. . 3 ⊢ ∪ ∪ ◡(◡𝐴 ∘ 𝐴) = (dom (◡𝐴 ∘ 𝐴) ∪ ran (◡𝐴 ∘ 𝐴))
7	5, 6	eqtr3i 2787	. 2 ⊢ ∪ ∪ (◡𝐴 ∘ 𝐴) = (dom (◡𝐴 ∘ 𝐴) ∪ ran (◡𝐴 ∘ 𝐴))
8		df-rn 5658	. . . . 5 ⊢ ran 𝐴 = dom ◡𝐴
9	8	eqcomi 2771	. . . 4 ⊢ dom ◡𝐴 = ran 𝐴
10		dmcoeq 5957	. . . 4 ⊢ (dom ◡𝐴 = ran 𝐴 → dom (◡𝐴 ∘ 𝐴) = dom 𝐴)
11	9, 10	ax-mp 5	. . 3 ⊢ dom (◡𝐴 ∘ 𝐴) = dom 𝐴
12		rncoeq 5958	. . . . 5 ⊢ (dom ◡𝐴 = ran 𝐴 → ran (◡𝐴 ∘ 𝐴) = ran ◡𝐴)
13	9, 12	ax-mp 5	. . . 4 ⊢ ran (◡𝐴 ∘ 𝐴) = ran ◡𝐴
14		dfdm4 5871	. . . 4 ⊢ dom 𝐴 = ran ◡𝐴
15	13, 14	eqtr4i 2788	. . 3 ⊢ ran (◡𝐴 ∘ 𝐴) = dom 𝐴
16	11, 15	uneq12i 4119	. 2 ⊢ (dom (◡𝐴 ∘ 𝐴) ∪ ran (◡𝐴 ∘ 𝐴)) = (dom 𝐴 ∪ dom 𝐴)
17		unidm 4110	. 2 ⊢ (dom 𝐴 ∪ dom 𝐴) = dom 𝐴
18	7, 16, 17	3eqtrri 2790	1 ⊢ dom 𝐴 = ∪ ∪ (◡𝐴 ∘ 𝐴)

Syntax hints:   = wceq 1560   ∪ cun 3902  ∪ cuni 4865  ^◡ccnv 5646  dom cdm 5647  ran crn 5648   ∘ ccom 5651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659

This theorem is referenced by: (None)