Theorem raldmqseu 39042

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 39040	. . 3 ⊢ (∀𝑢 ∈ (dom 𝑅 / 𝑅)∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅 ↔ ∀𝑢 ∈ (dom 𝑅 / 𝑅)∃!𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅)
2		ralrmo3 39041	. . 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 38918	. . . . . . 7 ⊢ ((𝑢 = [𝑡]𝑅 ∧ 𝑢 ∈ (dom 𝑅 / 𝑅)) ↔ (𝑢 = [𝑡]𝑅 ∧ [𝑡]𝑅 ∈ (dom 𝑅 / 𝑅)))
5		ancom 465	. . . . . . 7 ⊢ ((𝑢 = [𝑡]𝑅 ∧ 𝑢 ∈ (dom 𝑅 / 𝑅)) ↔ (𝑢 ∈ (dom 𝑅 / 𝑅) ∧ 𝑢 = [𝑡]𝑅))
6		ancom 465	. . . . . . 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 8783	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → ([𝑡]𝑅 ∈ (dom 𝑅 / 𝑅) ↔ 𝑡 ∈ dom 𝑅))
9	8	anbi1d 642	. . . . . 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 3383	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (∃*𝑡 ∈ dom 𝑅(𝑢 ∈ (dom 𝑅 / 𝑅) ∧ 𝑢 = [𝑡]𝑅) ↔ ∃*𝑡 ∈ dom 𝑅(𝑡 ∈ dom 𝑅 ∧ 𝑢 = [𝑡]𝑅)))
12		rmoanid 3378	. . . 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 1949	. 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 400  ∀wal 1567   = wceq 1569   ∈ wcel 2142  ∀wral 3078  ∃!wreu 3366  ∃*wrmo 3367  dom cdm 5660  [cec 8690   / cqs 8691

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  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 3368  df-reu 3369  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-xp 5666  df-rel 5667  df-cnv 5668  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-ec 8694  df-qs 8698

This theorem is used by:  disjqmap  39504