Theorem dfdm2 5322

Description: Alternate definition of domain df-dm 4784 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 4965	. . . . . 6 ⊢ ◡(◡𝐴 ∘ 𝐴) = (◡𝐴 ∘ ◡◡𝐴)
2		cocnvcnv2 5299	. . . . . 6 ⊢ (◡𝐴 ∘ ◡◡𝐴) = (◡𝐴 ∘ 𝐴)
3	1, 2	eqtri 2259	. . . . 5 ⊢ ◡(◡𝐴 ∘ 𝐴) = (◡𝐴 ∘ 𝐴)
4	3	unieqi 3945	. . . 4 ⊢ ∪ ◡(◡𝐴 ∘ 𝐴) = ∪ (◡𝐴 ∘ 𝐴)
5	4	unieqi 3945	. . 3 ⊢ ∪ ∪ ◡(◡𝐴 ∘ 𝐴) = ∪ ∪ (◡𝐴 ∘ 𝐴)
6		unidmrn 5320	. . 3 ⊢ ∪ ∪ ◡(◡𝐴 ∘ 𝐴) = (dom (◡𝐴 ∘ 𝐴) ∪ ran (◡𝐴 ∘ 𝐴))
7	5, 6	eqtr3i 2261	. 2 ⊢ ∪ ∪ (◡𝐴 ∘ 𝐴) = (dom (◡𝐴 ∘ 𝐴) ∪ ran (◡𝐴 ∘ 𝐴))
8		df-rn 4785	. . . . 5 ⊢ ran 𝐴 = dom ◡𝐴
9	8	eqcomi 2242	. . . 4 ⊢ dom ◡𝐴 = ran 𝐴
10		dmcoeq 5055	. . . 4 ⊢ (dom ◡𝐴 = ran 𝐴 → dom (◡𝐴 ∘ 𝐴) = dom 𝐴)
11	9, 10	ax-mp 5	. . 3 ⊢ dom (◡𝐴 ∘ 𝐴) = dom 𝐴
12		rncoeq 5056	. . . . 5 ⊢ (dom ◡𝐴 = ran 𝐴 → ran (◡𝐴 ∘ 𝐴) = ran ◡𝐴)
13	9, 12	ax-mp 5	. . . 4 ⊢ ran (◡𝐴 ∘ 𝐴) = ran ◡𝐴
14		dfdm4 4973	. . . 4 ⊢ dom 𝐴 = ran ◡𝐴
15	13, 14	eqtr4i 2262	. . 3 ⊢ ran (◡𝐴 ∘ 𝐴) = dom 𝐴
16	11, 15	uneq12i 3381	. 2 ⊢ (dom (◡𝐴 ∘ 𝐴) ∪ ran (◡𝐴 ∘ 𝐴)) = (dom 𝐴 ∪ dom 𝐴)
17		unidm 3372	. 2 ⊢ (dom 𝐴 ∪ dom 𝐴) = dom 𝐴
18	7, 16, 17	3eqtrri 2264	1 ⊢ dom 𝐴 = ∪ ∪ (◡𝐴 ∘ 𝐴)

This proof depends on syntax axioms:   = wceq 1402   ∪ cun 3218  ∪ cuni 3935  ^◡ccnv 4773  dom cdm 4774  ran crn 4775   ∘ ccom 4778

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786

This theorem is used by: (None)