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Theorem reldm 8034

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 8033	. . 3 ⊢ (Rel 𝐴 → (𝑦 ∈ dom 𝐴 ↔ ∃𝑧 ∈ 𝐴 (1^st ‘𝑧) = 𝑦))
2		fvex 6904	. . . . . 6 ⊢ (1^st ‘𝑥) ∈ V
3		eqid 2731	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥))
4	2, 3	fnmpti 6693	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥)) Fn 𝐴
5		fvelrnb 6952	. . . . 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 6891	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (1^st ‘𝑥) = (1^st ‘𝑧))
8		fvex 6904	. . . . . . . 8 ⊢ (1^st ‘𝑧) ∈ V
9	7, 3, 8	fvmpt 6998	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥))‘𝑧) = (1^st ‘𝑧))
10	9	eqeq1d 2733	. . . . . 6 ⊢ (𝑧 ∈ 𝐴 → (((𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥))‘𝑧) = 𝑦 ↔ (1^st ‘𝑧) = 𝑦))
11	10	rexbiia 3091	. . . . 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 2729	1 ⊢ (Rel 𝐴 → dom 𝐴 = ran (𝑥 ∈ 𝐴 ↦ (1^st ‘𝑥)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 = wceq 1540 ∈ wcel 2105 ∃wrex 3069 ↦ cmpt 5231 dom cdm 5676 ran crn 5677 Rel wrel 5681 Fn wfn 6538 ‘cfv 6543 1^st c1st 7977

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7729

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-iota 6495 df-fun 6545 df-fn 6546 df-fv 6551 df-1st 7979 df-2nd 7980

This theorem is referenced by: fidomdm 9335 dmct 10525

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