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Theorem reldm 8055

Description: An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.)

**Assertion**
Ref	Expression
reldm	⊢ (Rel 𝐴 → dom 𝐴 = ran (𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥)))

Distinct variable group: 𝑥,𝐴

Proof of Theorem reldm Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		releldm2 8054	. . 3 ⊢ (Rel 𝐴 → (𝑦 ∈ dom 𝐴 ↔ ∃𝑧 ∈ 𝐴 (1^st ‘𝑧) = 𝑦))
2		fvex 6898	. . . . . 6 ⊢ (1^st ‘𝑥) ∈ V
3		eqid 2761	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥))
4	2, 3	fnmpti 6682	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥)) Fn 𝐴
5		fvelrnb 6945	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥)) Fn 𝐴 → (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥)) ↔ ∃𝑧 ∈ 𝐴 ((𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥))‘𝑧) = 𝑦))
6	4, 5	ax-mp 5	. . . 4 ⊢ (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥)) ↔ ∃𝑧 ∈ 𝐴 ((𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥))‘𝑧) = 𝑦)
7		fveq2 6885	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (1^st ‘𝑥) = (1^st ‘𝑧))
8		fvex 6898	. . . . . . . 8 ⊢ (1^st ‘𝑧) ∈ V
9	7, 3, 8	fvmpt 6993	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥))‘𝑧) = (1^st ‘𝑧))
10	9	eqeq1d 2763	. . . . . 6 ⊢ (𝑧 ∈ 𝐴 → (((𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥))‘𝑧) = 𝑦 ↔ (1^st ‘𝑧) = 𝑦))
11	10	rexbiia 3108	. . . . 5 ⊢ (∃𝑧 ∈ 𝐴 ((𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥))‘𝑧) = 𝑦 ↔ ∃𝑧 ∈ 𝐴 (1^st ‘𝑧) = 𝑦)
12	11	a1i 11	. . . 4 ⊢ (Rel 𝐴 → (∃𝑧 ∈ 𝐴 ((𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥))‘𝑧) = 𝑦 ↔ ∃𝑧 ∈ 𝐴 (1^st ‘𝑧) = 𝑦))
13	6, 12	bitr2id 287	. . 3 ⊢ (Rel 𝐴 → (∃𝑧 ∈ 𝐴 (1^st ‘𝑧) = 𝑦 ↔ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥))))
14	1, 13	bitrd 282	. 2 ⊢ (Rel 𝐴 → (𝑦 ∈ dom 𝐴 ↔ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥))))
15	14	eqrdv 2759	1 ⊢ (Rel 𝐴 → dom 𝐴 = ran (𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 = wceq 1570 ∈ wcel 2145 ∃wrex 3087 ↦ cmpt 5186 dom cdm 5651 ran crn 5652 Rel wrel 5656 Fn wfn 6533 ‘cfv 6538 1^st c1st 7999

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-nul 5260 ax-pr 5391 ax-un 7751

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-rab 3414 df-v 3453 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-iota 6494 df-fun 6540 df-fn 6541 df-fv 6546 df-1st 8001 df-2nd 8002

This theorem is used by: fidomdm 9323 dmct 10602 dmctOLD 10603

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