Theorem dfdm2 5204

Description: Alternate definition of domain df-dm 4673 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 4851	. . . . . 6 ⊢ ◡(◡𝐴 ∘ 𝐴) = (◡𝐴 ∘ ◡◡𝐴)
2		cocnvcnv2 5181	. . . . . 6 ⊢ (◡𝐴 ∘ ◡◡𝐴) = (◡𝐴 ∘ 𝐴)
3	1, 2	eqtri 2217	. . . . 5 ⊢ ◡(◡𝐴 ∘ 𝐴) = (◡𝐴 ∘ 𝐴)
4	3	unieqi 3849	. . . 4 ⊢ ∪ ◡(◡𝐴 ∘ 𝐴) = ∪ (◡𝐴 ∘ 𝐴)
5	4	unieqi 3849	. . 3 ⊢ ∪ ∪ ◡(◡𝐴 ∘ 𝐴) = ∪ ∪ (◡𝐴 ∘ 𝐴)
6		unidmrn 5202	. . 3 ⊢ ∪ ∪ ◡(◡𝐴 ∘ 𝐴) = (dom (◡𝐴 ∘ 𝐴) ∪ ran (◡𝐴 ∘ 𝐴))
7	5, 6	eqtr3i 2219	. 2 ⊢ ∪ ∪ (◡𝐴 ∘ 𝐴) = (dom (◡𝐴 ∘ 𝐴) ∪ ran (◡𝐴 ∘ 𝐴))
8		df-rn 4674	. . . . 5 ⊢ ran 𝐴 = dom ◡𝐴
9	8	eqcomi 2200	. . . 4 ⊢ dom ◡𝐴 = ran 𝐴
10		dmcoeq 4938	. . . 4 ⊢ (dom ◡𝐴 = ran 𝐴 → dom (◡𝐴 ∘ 𝐴) = dom 𝐴)
11	9, 10	ax-mp 5	. . 3 ⊢ dom (◡𝐴 ∘ 𝐴) = dom 𝐴
12		rncoeq 4939	. . . . 5 ⊢ (dom ◡𝐴 = ran 𝐴 → ran (◡𝐴 ∘ 𝐴) = ran ◡𝐴)
13	9, 12	ax-mp 5	. . . 4 ⊢ ran (◡𝐴 ∘ 𝐴) = ran ◡𝐴
14		dfdm4 4858	. . . 4 ⊢ dom 𝐴 = ran ◡𝐴
15	13, 14	eqtr4i 2220	. . 3 ⊢ ran (◡𝐴 ∘ 𝐴) = dom 𝐴
16	11, 15	uneq12i 3315	. 2 ⊢ (dom (◡𝐴 ∘ 𝐴) ∪ ran (◡𝐴 ∘ 𝐴)) = (dom 𝐴 ∪ dom 𝐴)
17		unidm 3306	. 2 ⊢ (dom 𝐴 ∪ dom 𝐴) = dom 𝐴
18	7, 16, 17	3eqtrri 2222	1 ⊢ dom 𝐴 = ∪ ∪ (◡𝐴 ∘ 𝐴)

Syntax hints:   = wceq 1364   ∪ cun 3155  ∪ cuni 3839  ^◡ccnv 4662  dom cdm 4663  ran crn 4664   ∘ ccom 4667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675

This theorem is referenced by: (None)