dfdm2 - Metamath Proof Explorer

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Theorem dfdm2 6100

Description: Alternate definition of domain df-dm 5529 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 5720	. . . . . 6 ⊢ ^◡(^◡𝐴 ∘ 𝐴) = (^◡𝐴 ∘ ^◡^◡𝐴)
2		cocnvcnv2 6078	. . . . . 6 ⊢ (^◡𝐴 ∘ ^◡^◡𝐴) = (^◡𝐴 ∘ 𝐴)
3	1, 2	eqtri 2821	. . . . 5 ⊢ ^◡(^◡𝐴 ∘ 𝐴) = (^◡𝐴 ∘ 𝐴)
4	3	unieqi 4813	. . . 4 ⊢ ∪ ^◡(^◡𝐴 ∘ 𝐴) = ∪ (^◡𝐴 ∘ 𝐴)
5	4	unieqi 4813	. . 3 ⊢ ∪ ∪ ^◡(^◡𝐴 ∘ 𝐴) = ∪ ∪ (^◡𝐴 ∘ 𝐴)
6		unidmrn 6098	. . 3 ⊢ ∪ ∪ ^◡(^◡𝐴 ∘ 𝐴) = (dom (^◡𝐴 ∘ 𝐴) ∪ ran (^◡𝐴 ∘ 𝐴))
7	5, 6	eqtr3i 2823	. 2 ⊢ ∪ ∪ (^◡𝐴 ∘ 𝐴) = (dom (^◡𝐴 ∘ 𝐴) ∪ ran (^◡𝐴 ∘ 𝐴))
8		df-rn 5530	. . . . 5 ⊢ ran 𝐴 = dom ^◡𝐴
9	8	eqcomi 2807	. . . 4 ⊢ dom ^◡𝐴 = ran 𝐴
10		dmcoeq 5810	. . . 4 ⊢ (dom ^◡𝐴 = ran 𝐴 → dom (^◡𝐴 ∘ 𝐴) = dom 𝐴)
11	9, 10	ax-mp 5	. . 3 ⊢ dom (^◡𝐴 ∘ 𝐴) = dom 𝐴
12		rncoeq 5811	. . . . 5 ⊢ (dom ^◡𝐴 = ran 𝐴 → ran (^◡𝐴 ∘ 𝐴) = ran ^◡𝐴)
13	9, 12	ax-mp 5	. . . 4 ⊢ ran (^◡𝐴 ∘ 𝐴) = ran ^◡𝐴
14		dfdm4 5728	. . . 4 ⊢ dom 𝐴 = ran ^◡𝐴
15	13, 14	eqtr4i 2824	. . 3 ⊢ ran (^◡𝐴 ∘ 𝐴) = dom 𝐴
16	11, 15	uneq12i 4088	. 2 ⊢ (dom (^◡𝐴 ∘ 𝐴) ∪ ran (^◡𝐴 ∘ 𝐴)) = (dom 𝐴 ∪ dom 𝐴)
17		unidm 4079	. 2 ⊢ (dom 𝐴 ∪ dom 𝐴) = dom 𝐴
18	7, 16, 17	3eqtrri 2826	1 ⊢ dom 𝐴 = ∪ ∪ (^◡𝐴 ∘ 𝐴)

Colors of variables: wff setvar class

Syntax hints: = wceq 1538 ∪ cun 3879 ∪ cuni 4800 ^◡ccnv 5518 dom cdm 5519 ran crn 5520 ∘ ccom 5523

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531

This theorem is referenced by: (None)