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Theorem reldm 7452

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 7451	. . 3 ⊢ (Rel 𝐴 → (𝑦 ∈ dom 𝐴 ↔ ∃𝑧 ∈ 𝐴 (1^st ‘𝑧) = 𝑦))
2		fvex 6422	. . . . . 6 ⊢ (1^st ‘𝑥) ∈ V
3		eqid 2797	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥))
4	2, 3	fnmpti 6231	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥)) Fn 𝐴
5		fvelrnb 6466	. . . . 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 6409	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (1^st ‘𝑥) = (1^st ‘𝑧))
8		fvex 6422	. . . . . . . 8 ⊢ (1^st ‘𝑧) ∈ V
9	7, 3, 8	fvmpt 6505	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥))‘𝑧) = (1^st ‘𝑧))
10	9	eqeq1d 2799	. . . . . 6 ⊢ (𝑧 ∈ 𝐴 → (((𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥))‘𝑧) = 𝑦 ↔ (1^st ‘𝑧) = 𝑦))
11	10	rexbiia 3219	. . . . 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 276	. . 3 ⊢ (Rel 𝐴 → (∃𝑧 ∈ 𝐴 (1^st ‘𝑧) = 𝑦 ↔ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥))))
14	1, 13	bitrd 271	. 2 ⊢ (Rel 𝐴 → (𝑦 ∈ dom 𝐴 ↔ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥))))
15	14	eqrdv 2795	1 ⊢ (Rel 𝐴 → dom 𝐴 = ran (𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 = wceq 1653 ∈ wcel 2157 ∃wrex 3088 ↦ cmpt 4920 dom cdm 5310 ran crn 5311 Rel wrel 5315 Fn wfn 6094 ‘cfv 6099 1^st c1st 7397

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181

This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ral 3092 df-rex 3093 df-rab 3096 df-v 3385 df-sbc 3632 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-sn 4367 df-pr 4369 df-op 4373 df-uni 4627 df-br 4842 df-opab 4904 df-mpt 4921 df-id 5218 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-iota 6062 df-fun 6101 df-fn 6102 df-fv 6107 df-1st 7399 df-2nd 7400

This theorem is referenced by: fidomdm 8483 dmct 9632

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