dfdm2 - Metamath Proof Explorer

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Theorem dfdm2 6281

Description: Alternate definition of domain df-dm 5687 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 5886	. . . . . 6 ⊢ ^◡(^◡𝐴 ∘ 𝐴) = (^◡𝐴 ∘ ^◡^◡𝐴)
2		cocnvcnv2 6258	. . . . . 6 ⊢ (^◡𝐴 ∘ ^◡^◡𝐴) = (^◡𝐴 ∘ 𝐴)
3	1, 2	eqtri 2761	. . . . 5 ⊢ ^◡(^◡𝐴 ∘ 𝐴) = (^◡𝐴 ∘ 𝐴)
4	3	unieqi 4922	. . . 4 ⊢ ∪ ^◡(^◡𝐴 ∘ 𝐴) = ∪ (^◡𝐴 ∘ 𝐴)
5	4	unieqi 4922	. . 3 ⊢ ∪ ∪ ^◡(^◡𝐴 ∘ 𝐴) = ∪ ∪ (^◡𝐴 ∘ 𝐴)
6		unidmrn 6279	. . 3 ⊢ ∪ ∪ ^◡(^◡𝐴 ∘ 𝐴) = (dom (^◡𝐴 ∘ 𝐴) ∪ ran (^◡𝐴 ∘ 𝐴))
7	5, 6	eqtr3i 2763	. 2 ⊢ ∪ ∪ (^◡𝐴 ∘ 𝐴) = (dom (^◡𝐴 ∘ 𝐴) ∪ ran (^◡𝐴 ∘ 𝐴))
8		df-rn 5688	. . . . 5 ⊢ ran 𝐴 = dom ^◡𝐴
9	8	eqcomi 2742	. . . 4 ⊢ dom ^◡𝐴 = ran 𝐴
10		dmcoeq 5974	. . . 4 ⊢ (dom ^◡𝐴 = ran 𝐴 → dom (^◡𝐴 ∘ 𝐴) = dom 𝐴)
11	9, 10	ax-mp 5	. . 3 ⊢ dom (^◡𝐴 ∘ 𝐴) = dom 𝐴
12		rncoeq 5975	. . . . 5 ⊢ (dom ^◡𝐴 = ran 𝐴 → ran (^◡𝐴 ∘ 𝐴) = ran ^◡𝐴)
13	9, 12	ax-mp 5	. . . 4 ⊢ ran (^◡𝐴 ∘ 𝐴) = ran ^◡𝐴
14		dfdm4 5896	. . . 4 ⊢ dom 𝐴 = ran ^◡𝐴
15	13, 14	eqtr4i 2764	. . 3 ⊢ ran (^◡𝐴 ∘ 𝐴) = dom 𝐴
16	11, 15	uneq12i 4162	. 2 ⊢ (dom (^◡𝐴 ∘ 𝐴) ∪ ran (^◡𝐴 ∘ 𝐴)) = (dom 𝐴 ∪ dom 𝐴)
17		unidm 4153	. 2 ⊢ (dom 𝐴 ∪ dom 𝐴) = dom 𝐴
18	7, 16, 17	3eqtrri 2766	1 ⊢ dom 𝐴 = ∪ ∪ (^◡𝐴 ∘ 𝐴)

Colors of variables: wff setvar class

Syntax hints: = wceq 1542 ∪ cun 3947 ∪ cuni 4909 ^◡ccnv 5676 dom cdm 5677 ran crn 5678 ∘ ccom 5681

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689

This theorem is referenced by: (None)