dfdm2 - Metamath Proof Explorer

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Theorem dfdm2 6312

Description: Alternate definition of domain df-dm 5710 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 5910	. . . . . 6 ⊢ ^◡(^◡𝐴 ∘ 𝐴) = (^◡𝐴 ∘ ^◡^◡𝐴)
2		cocnvcnv2 6289	. . . . . 6 ⊢ (^◡𝐴 ∘ ^◡^◡𝐴) = (^◡𝐴 ∘ 𝐴)
3	1, 2	eqtri 2768	. . . . 5 ⊢ ^◡(^◡𝐴 ∘ 𝐴) = (^◡𝐴 ∘ 𝐴)
4	3	unieqi 4943	. . . 4 ⊢ ∪ ^◡(^◡𝐴 ∘ 𝐴) = ∪ (^◡𝐴 ∘ 𝐴)
5	4	unieqi 4943	. . 3 ⊢ ∪ ∪ ^◡(^◡𝐴 ∘ 𝐴) = ∪ ∪ (^◡𝐴 ∘ 𝐴)
6		unidmrn 6310	. . 3 ⊢ ∪ ∪ ^◡(^◡𝐴 ∘ 𝐴) = (dom (^◡𝐴 ∘ 𝐴) ∪ ran (^◡𝐴 ∘ 𝐴))
7	5, 6	eqtr3i 2770	. 2 ⊢ ∪ ∪ (^◡𝐴 ∘ 𝐴) = (dom (^◡𝐴 ∘ 𝐴) ∪ ran (^◡𝐴 ∘ 𝐴))
8		df-rn 5711	. . . . 5 ⊢ ran 𝐴 = dom ^◡𝐴
9	8	eqcomi 2749	. . . 4 ⊢ dom ^◡𝐴 = ran 𝐴
10		dmcoeq 6001	. . . 4 ⊢ (dom ^◡𝐴 = ran 𝐴 → dom (^◡𝐴 ∘ 𝐴) = dom 𝐴)
11	9, 10	ax-mp 5	. . 3 ⊢ dom (^◡𝐴 ∘ 𝐴) = dom 𝐴
12		rncoeq 6002	. . . . 5 ⊢ (dom ^◡𝐴 = ran 𝐴 → ran (^◡𝐴 ∘ 𝐴) = ran ^◡𝐴)
13	9, 12	ax-mp 5	. . . 4 ⊢ ran (^◡𝐴 ∘ 𝐴) = ran ^◡𝐴
14		dfdm4 5920	. . . 4 ⊢ dom 𝐴 = ran ^◡𝐴
15	13, 14	eqtr4i 2771	. . 3 ⊢ ran (^◡𝐴 ∘ 𝐴) = dom 𝐴
16	11, 15	uneq12i 4189	. 2 ⊢ (dom (^◡𝐴 ∘ 𝐴) ∪ ran (^◡𝐴 ∘ 𝐴)) = (dom 𝐴 ∪ dom 𝐴)
17		unidm 4180	. 2 ⊢ (dom 𝐴 ∪ dom 𝐴) = dom 𝐴
18	7, 16, 17	3eqtrri 2773	1 ⊢ dom 𝐴 = ∪ ∪ (^◡𝐴 ∘ 𝐴)

Colors of variables: wff setvar class

Syntax hints: = wceq 1537 ∪ cun 3974 ∪ cuni 4931 ^◡ccnv 5699 dom cdm 5700 ran crn 5701 ∘ ccom 5704

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-clab 2718 df-cleq 2732 df-clel 2819 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712

This theorem is referenced by: (None)