Theorem reldm 6077

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 ↦ (1st ‘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 6076	. . 3 ⊢ (Rel 𝐴 → (𝑦 ∈ dom 𝐴 ↔ ∃𝑧 ∈ 𝐴 (1st ‘𝑧) = 𝑦))
2		vex 2684	. . . . . . 7 ⊢ 𝑥 ∈ V
3		1stexg 6058	. . . . . . 7 ⊢ (𝑥 ∈ V → (1st ‘𝑥) ∈ V)
4	2, 3	ax-mp 5	. . . . . 6 ⊢ (1st ‘𝑥) ∈ V
5		eqid 2137	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥))
6	4, 5	fnmpti 5246	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥)) Fn 𝐴
7		fvelrnb 5462	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↦ (1st ‘𝑥)) Fn 𝐴 → (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥)) ↔ ∃𝑧 ∈ 𝐴 ((𝑥 ∈ 𝐴 ↦ (1st ‘𝑥))‘𝑧) = 𝑦))
8	6, 7	ax-mp 5	. . . 4 ⊢ (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥)) ↔ ∃𝑧 ∈ 𝐴 ((𝑥 ∈ 𝐴 ↦ (1st ‘𝑥))‘𝑧) = 𝑦)
9		fveq2 5414	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (1st ‘𝑥) = (1st ‘𝑧))
10		vex 2684	. . . . . . . . 9 ⊢ 𝑧 ∈ V
11		1stexg 6058	. . . . . . . . 9 ⊢ (𝑧 ∈ V → (1st ‘𝑧) ∈ V)
12	10, 11	ax-mp 5	. . . . . . . 8 ⊢ (1st ‘𝑧) ∈ V
13	9, 5, 12	fvmpt 5491	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (1st ‘𝑥))‘𝑧) = (1st ‘𝑧))
14	13	eqeq1d 2146	. . . . . 6 ⊢ (𝑧 ∈ 𝐴 → (((𝑥 ∈ 𝐴 ↦ (1st ‘𝑥))‘𝑧) = 𝑦 ↔ (1st ‘𝑧) = 𝑦))
15	14	rexbiia 2448	. . . . 5 ⊢ (∃𝑧 ∈ 𝐴 ((𝑥 ∈ 𝐴 ↦ (1st ‘𝑥))‘𝑧) = 𝑦 ↔ ∃𝑧 ∈ 𝐴 (1st ‘𝑧) = 𝑦)
16	15	a1i 9	. . . 4 ⊢ (Rel 𝐴 → (∃𝑧 ∈ 𝐴 ((𝑥 ∈ 𝐴 ↦ (1st ‘𝑥))‘𝑧) = 𝑦 ↔ ∃𝑧 ∈ 𝐴 (1st ‘𝑧) = 𝑦))
17	8, 16	syl5rbb 192	. . 3 ⊢ (Rel 𝐴 → (∃𝑧 ∈ 𝐴 (1st ‘𝑧) = 𝑦 ↔ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥))))
18	1, 17	bitrd 187	. 2 ⊢ (Rel 𝐴 → (𝑦 ∈ dom 𝐴 ↔ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥))))
19	18	eqrdv 2135	1 ⊢ (Rel 𝐴 → dom 𝐴 = ran (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1331   ∈ wcel 1480  ∃wrex 2415  Vcvv 2681   ↦ cmpt 3984  dom cdm 4534  ran crn 4535  Rel wrel 4539   Fn wfn 5113  ‘cfv 5118  1^st c1st 6029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fo 5124  df-fv 5126  df-1st 6031  df-2nd 6032

This theorem is referenced by: (None)