Theorem reldm 6165

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 6164	. . 3 ⊢ (Rel 𝐴 → (𝑦 ∈ dom 𝐴 ↔ ∃𝑧 ∈ 𝐴 (1st ‘𝑧) = 𝑦))
2		vex 2733	. . . . . . 7 ⊢ 𝑥 ∈ V
3		1stexg 6146	. . . . . . 7 ⊢ (𝑥 ∈ V → (1st ‘𝑥) ∈ V)
4	2, 3	ax-mp 5	. . . . . 6 ⊢ (1st ‘𝑥) ∈ V
5		eqid 2170	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥))
6	4, 5	fnmpti 5326	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥)) Fn 𝐴
7		fvelrnb 5544	. . . . 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 5496	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (1st ‘𝑥) = (1st ‘𝑧))
10		vex 2733	. . . . . . . . 9 ⊢ 𝑧 ∈ V
11		1stexg 6146	. . . . . . . . 9 ⊢ (𝑧 ∈ V → (1st ‘𝑧) ∈ V)
12	10, 11	ax-mp 5	. . . . . . . 8 ⊢ (1st ‘𝑧) ∈ V
13	9, 5, 12	fvmpt 5573	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (1st ‘𝑥))‘𝑧) = (1st ‘𝑧))
14	13	eqeq1d 2179	. . . . . 6 ⊢ (𝑧 ∈ 𝐴 → (((𝑥 ∈ 𝐴 ↦ (1st ‘𝑥))‘𝑧) = 𝑦 ↔ (1st ‘𝑧) = 𝑦))
15	14	rexbiia 2485	. . . . 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	bitr2id 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 2168	1 ⊢ (Rel 𝐴 → dom 𝐴 = ran (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1348   ∈ wcel 2141  ∃wrex 2449  Vcvv 2730   ↦ cmpt 4050  dom cdm 4611  ran crn 4612  Rel wrel 4616   Fn wfn 5193  ‘cfv 5198  1^st c1st 6117

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fo 5204  df-fv 5206  df-1st 6119  df-2nd 6120

This theorem is referenced by: (None)