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Theorem reldm 7219

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 7218	. . 3 ⊢ (Rel 𝐴 → (𝑦 ∈ dom 𝐴 ↔ ∃𝑧 ∈ 𝐴 (1^st ‘𝑧) = 𝑦))
2		fvex 6201	. . . . . 6 ⊢ (1^st ‘𝑥) ∈ V
3		eqid 2622	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥))
4	2, 3	fnmpti 6022	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥)) Fn 𝐴
5		fvelrnb 6243	. . . . 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 6191	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (1^st ‘𝑥) = (1^st ‘𝑧))
8		fvex 6201	. . . . . . . 8 ⊢ (1^st ‘𝑧) ∈ V
9	7, 3, 8	fvmpt 6282	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥))‘𝑧) = (1^st ‘𝑧))
10	9	eqeq1d 2624	. . . . . 6 ⊢ (𝑧 ∈ 𝐴 → (((𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥))‘𝑧) = 𝑦 ↔ (1^st ‘𝑧) = 𝑦))
11	10	rexbiia 3040	. . . . 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	syl5rbb 273	. . 3 ⊢ (Rel 𝐴 → (∃𝑧 ∈ 𝐴 (1^st ‘𝑧) = 𝑦 ↔ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥))))
14	1, 13	bitrd 268	. 2 ⊢ (Rel 𝐴 → (𝑦 ∈ dom 𝐴 ↔ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥))))
15	14	eqrdv 2620	1 ⊢ (Rel 𝐴 → dom 𝐴 = ran (𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 ↦ cmpt 4729 dom cdm 5114 ran crn 5115 Rel wrel 5119 Fn wfn 5883 ‘cfv 5888 1^st c1st 7166

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-fv 5896 df-1st 7168 df-2nd 7169

This theorem is referenced by: fidomdm 8243 dmct 9346

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