Theorem reldm 8065

Description: An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.) (Proof shortened by Sky Winter, 22-Oct-2026.)

Assertion

Ref	Expression
reldm	⊢ (Rel 𝐴 → dom 𝐴 = ran (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥)))

Distinct variable group:   𝑥,𝐴

Proof of Theorem reldm

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		releldm2 8064	. . 3 ⊢ (Rel 𝐴 → (𝑦 ∈ dom 𝐴 ↔ ∃𝑥 ∈ 𝐴 (1st ‘𝑥) = 𝑦))
2		eqid 2763	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥))
3		fvex 6909	. . . . 5 ⊢ (1st ‘𝑥) ∈ V
4	2, 3	elrnmpti 5949	. . . 4 ⊢ (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥)) ↔ ∃𝑥 ∈ 𝐴 𝑦 = (1st ‘𝑥))
5		eqcom 2770	. . . . 5 ⊢ (𝑦 = (1st ‘𝑥) ↔ (1st ‘𝑥) = 𝑦)
6	5	rexbii 3113	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = (1st ‘𝑥) ↔ ∃𝑥 ∈ 𝐴 (1st ‘𝑥) = 𝑦)
7	4, 6	bitri 278	. . 3 ⊢ (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥)) ↔ ∃𝑥 ∈ 𝐴 (1st ‘𝑥) = 𝑦)
8	1, 7	bitr4di 292	. 2 ⊢ (Rel 𝐴 → (𝑦 ∈ dom 𝐴 ↔ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥))))
9	8	eqrdv 2761	1 ⊢ (Rel 𝐴 → dom 𝐴 = ran (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥)))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2146  ∃wrex 3090   ↦ cmpt 5189  dom cdm 5655  ran crn 5656  Rel wrel 5660  ‘cfv 6547  1^st c1st 8009

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7761

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6501  df-fun 6549  df-fv 6555  df-1st 8011  df-2nd 8012

This theorem is used by:  fidomdm  9337  dmct  10615  dmctOLD  10616