dfdm2 - Metamath Proof Explorer

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Theorem dfdm2 6257

Description: Alternate definition of domain df-dm 5651 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 5852	. . . . . 6 ⊢ ^◡(^◡𝐴 ∘ 𝐴) = (^◡𝐴 ∘ ^◡^◡𝐴)
2		cocnvcnv2 6234	. . . . . 6 ⊢ (^◡𝐴 ∘ ^◡^◡𝐴) = (^◡𝐴 ∘ 𝐴)
3	1, 2	eqtri 2753	. . . . 5 ⊢ ^◡(^◡𝐴 ∘ 𝐴) = (^◡𝐴 ∘ 𝐴)
4	3	unieqi 4886	. . . 4 ⊢ ∪ ^◡(^◡𝐴 ∘ 𝐴) = ∪ (^◡𝐴 ∘ 𝐴)
5	4	unieqi 4886	. . 3 ⊢ ∪ ∪ ^◡(^◡𝐴 ∘ 𝐴) = ∪ ∪ (^◡𝐴 ∘ 𝐴)
6		unidmrn 6255	. . 3 ⊢ ∪ ∪ ^◡(^◡𝐴 ∘ 𝐴) = (dom (^◡𝐴 ∘ 𝐴) ∪ ran (^◡𝐴 ∘ 𝐴))
7	5, 6	eqtr3i 2755	. 2 ⊢ ∪ ∪ (^◡𝐴 ∘ 𝐴) = (dom (^◡𝐴 ∘ 𝐴) ∪ ran (^◡𝐴 ∘ 𝐴))
8		df-rn 5652	. . . . 5 ⊢ ran 𝐴 = dom ^◡𝐴
9	8	eqcomi 2739	. . . 4 ⊢ dom ^◡𝐴 = ran 𝐴
10		dmcoeq 5945	. . . 4 ⊢ (dom ^◡𝐴 = ran 𝐴 → dom (^◡𝐴 ∘ 𝐴) = dom 𝐴)
11	9, 10	ax-mp 5	. . 3 ⊢ dom (^◡𝐴 ∘ 𝐴) = dom 𝐴
12		rncoeq 5946	. . . . 5 ⊢ (dom ^◡𝐴 = ran 𝐴 → ran (^◡𝐴 ∘ 𝐴) = ran ^◡𝐴)
13	9, 12	ax-mp 5	. . . 4 ⊢ ran (^◡𝐴 ∘ 𝐴) = ran ^◡𝐴
14		dfdm4 5862	. . . 4 ⊢ dom 𝐴 = ran ^◡𝐴
15	13, 14	eqtr4i 2756	. . 3 ⊢ ran (^◡𝐴 ∘ 𝐴) = dom 𝐴
16	11, 15	uneq12i 4132	. 2 ⊢ (dom (^◡𝐴 ∘ 𝐴) ∪ ran (^◡𝐴 ∘ 𝐴)) = (dom 𝐴 ∪ dom 𝐴)
17		unidm 4123	. 2 ⊢ (dom 𝐴 ∪ dom 𝐴) = dom 𝐴
18	7, 16, 17	3eqtrri 2758	1 ⊢ dom 𝐴 = ∪ ∪ (^◡𝐴 ∘ 𝐴)

Colors of variables: wff setvar class

Syntax hints: = wceq 1540 ∪ cun 3915 ∪ cuni 4874 ^◡ccnv 5640 dom cdm 5641 ran crn 5642 ∘ ccom 5645

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 2702 ax-sep 5254 ax-nul 5264 ax-pr 5390

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 2709 df-cleq 2722 df-clel 2804 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-br 5111 df-opab 5173 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653

This theorem is referenced by: (None)