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Theorem reldm 7979

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 7978	. . 3 ⊢ (Rel 𝐴 → (𝑦 ∈ dom 𝐴 ↔ ∃𝑧 ∈ 𝐴 (1^st ‘𝑧) = 𝑦))
2		fvex 6835	. . . . . 6 ⊢ (1^st ‘𝑥) ∈ V
3		eqid 2729	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥))
4	2, 3	fnmpti 6625	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥)) Fn 𝐴
5		fvelrnb 6883	. . . . 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 6822	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (1^st ‘𝑥) = (1^st ‘𝑧))
8		fvex 6835	. . . . . . . 8 ⊢ (1^st ‘𝑧) ∈ V
9	7, 3, 8	fvmpt 6930	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥))‘𝑧) = (1^st ‘𝑧))
10	9	eqeq1d 2731	. . . . . 6 ⊢ (𝑧 ∈ 𝐴 → (((𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥))‘𝑧) = 𝑦 ↔ (1^st ‘𝑧) = 𝑦))
11	10	rexbiia 3074	. . . . 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 284	. . 3 ⊢ (Rel 𝐴 → (∃𝑧 ∈ 𝐴 (1^st ‘𝑧) = 𝑦 ↔ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥))))
14	1, 13	bitrd 279	. 2 ⊢ (Rel 𝐴 → (𝑦 ∈ dom 𝐴 ↔ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥))))
15	14	eqrdv 2727	1 ⊢ (Rel 𝐴 → dom 𝐴 = ran (𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2109 ∃wrex 3053 ↦ cmpt 5173 dom cdm 5619 ran crn 5620 Rel wrel 5624 Fn wfn 6477 ‘cfv 6482 1^st c1st 7922

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-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5235 ax-nul 5245 ax-pr 5371 ax-un 7671

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-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3395 df-v 3438 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4285 df-if 4477 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-br 5093 df-opab 5155 df-mpt 5174 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-iota 6438 df-fun 6484 df-fn 6485 df-fv 6490 df-1st 7924 df-2nd 7925

This theorem is referenced by: fidomdm 9224 dmct 10418

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