Theorem reldm 6204

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 6203	. . 3 ⊢ (Rel 𝐴 → (𝑦 ∈ dom 𝐴 ↔ ∃𝑧 ∈ 𝐴 (1st ‘𝑧) = 𝑦))
2		vex 2754	. . . . . . 7 ⊢ 𝑥 ∈ V
3		1stexg 6185	. . . . . . 7 ⊢ (𝑥 ∈ V → (1st ‘𝑥) ∈ V)
4	2, 3	ax-mp 5	. . . . . 6 ⊢ (1st ‘𝑥) ∈ V
5		eqid 2188	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥))
6	4, 5	fnmpti 5358	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥)) Fn 𝐴
7		fvelrnb 5578	. . . . 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 5529	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (1st ‘𝑥) = (1st ‘𝑧))
10		vex 2754	. . . . . . . . 9 ⊢ 𝑧 ∈ V
11		1stexg 6185	. . . . . . . . 9 ⊢ (𝑧 ∈ V → (1st ‘𝑧) ∈ V)
12	10, 11	ax-mp 5	. . . . . . . 8 ⊢ (1st ‘𝑧) ∈ V
13	9, 5, 12	fvmpt 5608	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (1st ‘𝑥))‘𝑧) = (1st ‘𝑧))
14	13	eqeq1d 2197	. . . . . 6 ⊢ (𝑧 ∈ 𝐴 → (((𝑥 ∈ 𝐴 ↦ (1st ‘𝑥))‘𝑧) = 𝑦 ↔ (1st ‘𝑧) = 𝑦))
15	14	rexbiia 2504	. . . . 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 193	. . 3 ⊢ (Rel 𝐴 → (∃𝑧 ∈ 𝐴 (1st ‘𝑧) = 𝑦 ↔ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥))))
18	1, 17	bitrd 188	. 2 ⊢ (Rel 𝐴 → (𝑦 ∈ dom 𝐴 ↔ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥))))
19	18	eqrdv 2186	1 ⊢ (Rel 𝐴 → dom 𝐴 = ran (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1363   ∈ wcel 2159  ∃wrex 2468  Vcvv 2751   ↦ cmpt 4078  dom cdm 4640  ran crn 4641  Rel wrel 4645   Fn wfn 5225  ‘cfv 5230  1^st c1st 6156

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-pow 4188  ax-pr 4223  ax-un 4447

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ral 2472  df-rex 2473  df-v 2753  df-sbc 2977  df-un 3147  df-in 3149  df-ss 3156  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-br 4018  df-opab 4079  df-mpt 4080  df-id 4307  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-fo 5236  df-fv 5238  df-1st 6158  df-2nd 6159

This theorem is referenced by: (None)