dfdm2 - Metamath Proof Explorer

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Theorem dfdm2 6254

Description: Alternate definition of domain df-dm 5648 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 5849	. . . . . 6 ⊢ ^◡(^◡𝐴 ∘ 𝐴) = (^◡𝐴 ∘ ^◡^◡𝐴)
2		cocnvcnv2 6231	. . . . . 6 ⊢ (^◡𝐴 ∘ ^◡^◡𝐴) = (^◡𝐴 ∘ 𝐴)
3	1, 2	eqtri 2752	. . . . 5 ⊢ ^◡(^◡𝐴 ∘ 𝐴) = (^◡𝐴 ∘ 𝐴)
4	3	unieqi 4883	. . . 4 ⊢ ∪ ^◡(^◡𝐴 ∘ 𝐴) = ∪ (^◡𝐴 ∘ 𝐴)
5	4	unieqi 4883	. . 3 ⊢ ∪ ∪ ^◡(^◡𝐴 ∘ 𝐴) = ∪ ∪ (^◡𝐴 ∘ 𝐴)
6		unidmrn 6252	. . 3 ⊢ ∪ ∪ ^◡(^◡𝐴 ∘ 𝐴) = (dom (^◡𝐴 ∘ 𝐴) ∪ ran (^◡𝐴 ∘ 𝐴))
7	5, 6	eqtr3i 2754	. 2 ⊢ ∪ ∪ (^◡𝐴 ∘ 𝐴) = (dom (^◡𝐴 ∘ 𝐴) ∪ ran (^◡𝐴 ∘ 𝐴))
8		df-rn 5649	. . . . 5 ⊢ ran 𝐴 = dom ^◡𝐴
9	8	eqcomi 2738	. . . 4 ⊢ dom ^◡𝐴 = ran 𝐴
10		dmcoeq 5942	. . . 4 ⊢ (dom ^◡𝐴 = ran 𝐴 → dom (^◡𝐴 ∘ 𝐴) = dom 𝐴)
11	9, 10	ax-mp 5	. . 3 ⊢ dom (^◡𝐴 ∘ 𝐴) = dom 𝐴
12		rncoeq 5943	. . . . 5 ⊢ (dom ^◡𝐴 = ran 𝐴 → ran (^◡𝐴 ∘ 𝐴) = ran ^◡𝐴)
13	9, 12	ax-mp 5	. . . 4 ⊢ ran (^◡𝐴 ∘ 𝐴) = ran ^◡𝐴
14		dfdm4 5859	. . . 4 ⊢ dom 𝐴 = ran ^◡𝐴
15	13, 14	eqtr4i 2755	. . 3 ⊢ ran (^◡𝐴 ∘ 𝐴) = dom 𝐴
16	11, 15	uneq12i 4129	. 2 ⊢ (dom (^◡𝐴 ∘ 𝐴) ∪ ran (^◡𝐴 ∘ 𝐴)) = (dom 𝐴 ∪ dom 𝐴)
17		unidm 4120	. 2 ⊢ (dom 𝐴 ∪ dom 𝐴) = dom 𝐴
18	7, 16, 17	3eqtrri 2757	1 ⊢ dom 𝐴 = ∪ ∪ (^◡𝐴 ∘ 𝐴)

Colors of variables: wff setvar class

Syntax hints: = wceq 1540 ∪ cun 3912 ∪ cuni 4871 ^◡ccnv 5637 dom cdm 5638 ran crn 5639 ∘ ccom 5642

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pr 5387

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-clab 2708 df-cleq 2721 df-clel 2803 df-ral 3045 df-rex 3054 df-rab 3406 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-br 5108 df-opab 5170 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650

This theorem is referenced by: (None)