Theorem raldmqseu 39100

Description: Equivalence between "exactly one" on the quotient carrier and "at most one" globally. Provides a type-safe way to talk about unique representatives either as ∃! on the intended carrier or as a global ∃* statement. (Contributed by Peter Mazsa, 6-Feb-2026.)

Assertion

Ref	Expression
raldmqseu	⊢ (𝑅 ∈ 𝑉 → (∀𝑢 ∈ (dom 𝑅 / 𝑅)∃!𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅 ↔ ∀𝑢∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅))

Distinct variable groups:   𝑡,𝑅,𝑢   𝑡,𝑉,𝑢

Proof of Theorem raldmqseu

Step	Hyp	Ref	Expression
1		raldmqsmo 39098	. . 3 ⊢ (∀𝑢 ∈ (dom 𝑅 / 𝑅)∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅 ↔ ∀𝑢 ∈ (dom 𝑅 / 𝑅)∃!𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅)
2		ralrmo3 39099	. . 3 ⊢ (∀𝑢 ∈ (dom 𝑅 / 𝑅)∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅 ↔ ∀𝑢∃*𝑡 ∈ dom 𝑅(𝑢 ∈ (dom 𝑅 / 𝑅) ∧ 𝑢 = [𝑡]𝑅))
3	1, 2	bitr3i 280	. 2 ⊢ (∀𝑢 ∈ (dom 𝑅 / 𝑅)∃!𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅 ↔ ∀𝑢∃*𝑡 ∈ dom 𝑅(𝑢 ∈ (dom 𝑅 / 𝑅) ∧ 𝑢 = [𝑡]𝑅))
4		eqelb 38976	. . . . . . 7 ⊢ ((𝑢 = [𝑡]𝑅 ∧ 𝑢 ∈ (dom 𝑅 / 𝑅)) ↔ (𝑢 = [𝑡]𝑅 ∧ [𝑡]𝑅 ∈ (dom 𝑅 / 𝑅)))
5		ancom 466	. . . . . . 7 ⊢ ((𝑢 = [𝑡]𝑅 ∧ 𝑢 ∈ (dom 𝑅 / 𝑅)) ↔ (𝑢 ∈ (dom 𝑅 / 𝑅) ∧ 𝑢 = [𝑡]𝑅))
6		ancom 466	. . . . . . 7 ⊢ ((𝑢 = [𝑡]𝑅 ∧ [𝑡]𝑅 ∈ (dom 𝑅 / 𝑅)) ↔ ([𝑡]𝑅 ∈ (dom 𝑅 / 𝑅) ∧ 𝑢 = [𝑡]𝑅))
7	4, 5, 6	3bitr3i 304	. . . . . 6 ⊢ ((𝑢 ∈ (dom 𝑅 / 𝑅) ∧ 𝑢 = [𝑡]𝑅) ↔ ([𝑡]𝑅 ∈ (dom 𝑅 / 𝑅) ∧ 𝑢 = [𝑡]𝑅))
8		eceldmqs 8790	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → ([𝑡]𝑅 ∈ (dom 𝑅 / 𝑅) ↔ 𝑡 ∈ dom 𝑅))
9	8	anbi1d 643	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (([𝑡]𝑅 ∈ (dom 𝑅 / 𝑅) ∧ 𝑢 = [𝑡]𝑅) ↔ (𝑡 ∈ dom 𝑅 ∧ 𝑢 = [𝑡]𝑅)))
10	7, 9	bitrid 286	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → ((𝑢 ∈ (dom 𝑅 / 𝑅) ∧ 𝑢 = [𝑡]𝑅) ↔ (𝑡 ∈ dom 𝑅 ∧ 𝑢 = [𝑡]𝑅)))
11	10	rmobidv 3382	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (∃*𝑡 ∈ dom 𝑅(𝑢 ∈ (dom 𝑅 / 𝑅) ∧ 𝑢 = [𝑡]𝑅) ↔ ∃*𝑡 ∈ dom 𝑅(𝑡 ∈ dom 𝑅 ∧ 𝑢 = [𝑡]𝑅)))
12		rmoanid 3377	. . . 4 ⊢ (∃*𝑡 ∈ dom 𝑅(𝑡 ∈ dom 𝑅 ∧ 𝑢 = [𝑡]𝑅) ↔ ∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅)
13	11, 12	bitrdi 290	. . 3 ⊢ (𝑅 ∈ 𝑉 → (∃*𝑡 ∈ dom 𝑅(𝑢 ∈ (dom 𝑅 / 𝑅) ∧ 𝑢 = [𝑡]𝑅) ↔ ∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅))
14	13	albidv 1953	. 2 ⊢ (𝑅 ∈ 𝑉 → (∀𝑢∃*𝑡 ∈ dom 𝑅(𝑢 ∈ (dom 𝑅 / 𝑅) ∧ 𝑢 = [𝑡]𝑅) ↔ ∀𝑢∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅))
15	3, 14	bitrid 286	1 ⊢ (𝑅 ∈ 𝑉 → (∀𝑢 ∈ (dom 𝑅 / 𝑅)∃!𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅 ↔ ∀𝑢∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ∃!wreu 3365  ∃*wrmo 3366  dom cdm 5659  [cec 8697   / cqs 8698

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ec 8701  df-qs 8705

This theorem is used by:  disjqmap  39562