Theorem raldmqseu 38535

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 38533	. . 3 ⊢ (∀𝑢 ∈ (dom 𝑅 / 𝑅)∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅 ↔ ∀𝑢 ∈ (dom 𝑅 / 𝑅)∃!𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅)
2		ralrmo3 38534	. . 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 277	. 2 ⊢ (∀𝑢 ∈ (dom 𝑅 / 𝑅)∃!𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅 ↔ ∀𝑢∃*𝑡 ∈ dom 𝑅(𝑢 ∈ (dom 𝑅 / 𝑅) ∧ 𝑢 = [𝑡]𝑅))
4		eqelb 38411	. . . . . . 7 ⊢ ((𝑢 = [𝑡]𝑅 ∧ 𝑢 ∈ (dom 𝑅 / 𝑅)) ↔ (𝑢 = [𝑡]𝑅 ∧ [𝑡]𝑅 ∈ (dom 𝑅 / 𝑅)))
5		ancom 460	. . . . . . 7 ⊢ ((𝑢 = [𝑡]𝑅 ∧ 𝑢 ∈ (dom 𝑅 / 𝑅)) ↔ (𝑢 ∈ (dom 𝑅 / 𝑅) ∧ 𝑢 = [𝑡]𝑅))
6		ancom 460	. . . . . . 7 ⊢ ((𝑢 = [𝑡]𝑅 ∧ [𝑡]𝑅 ∈ (dom 𝑅 / 𝑅)) ↔ ([𝑡]𝑅 ∈ (dom 𝑅 / 𝑅) ∧ 𝑢 = [𝑡]𝑅))
7	4, 5, 6	3bitr3i 301	. . . . . 6 ⊢ ((𝑢 ∈ (dom 𝑅 / 𝑅) ∧ 𝑢 = [𝑡]𝑅) ↔ ([𝑡]𝑅 ∈ (dom 𝑅 / 𝑅) ∧ 𝑢 = [𝑡]𝑅))
8		eceldmqs 8726	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → ([𝑡]𝑅 ∈ (dom 𝑅 / 𝑅) ↔ 𝑡 ∈ dom 𝑅))
9	8	anbi1d 632	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (([𝑡]𝑅 ∈ (dom 𝑅 / 𝑅) ∧ 𝑢 = [𝑡]𝑅) ↔ (𝑡 ∈ dom 𝑅 ∧ 𝑢 = [𝑡]𝑅)))
10	7, 9	bitrid 283	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → ((𝑢 ∈ (dom 𝑅 / 𝑅) ∧ 𝑢 = [𝑡]𝑅) ↔ (𝑡 ∈ dom 𝑅 ∧ 𝑢 = [𝑡]𝑅)))
11	10	rmobidv 3364	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (∃*𝑡 ∈ dom 𝑅(𝑢 ∈ (dom 𝑅 / 𝑅) ∧ 𝑢 = [𝑡]𝑅) ↔ ∃*𝑡 ∈ dom 𝑅(𝑡 ∈ dom 𝑅 ∧ 𝑢 = [𝑡]𝑅)))
12		rmoanid 3359	. . . 4 ⊢ (∃*𝑡 ∈ dom 𝑅(𝑡 ∈ dom 𝑅 ∧ 𝑢 = [𝑡]𝑅) ↔ ∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅)
13	11, 12	bitrdi 287	. . 3 ⊢ (𝑅 ∈ 𝑉 → (∃*𝑡 ∈ dom 𝑅(𝑢 ∈ (dom 𝑅 / 𝑅) ∧ 𝑢 = [𝑡]𝑅) ↔ ∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅))
14	13	albidv 1922	. 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 283	1 ⊢ (𝑅 ∈ 𝑉 → (∀𝑢 ∈ (dom 𝑅 / 𝑅)∃!𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅 ↔ ∀𝑢∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   = wceq 1542   ∈ wcel 2114  ∀wral 3050  ∃!wreu 3347  ∃*wrmo 3348  dom cdm 5623  [cec 8633   / cqs 8634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-xp 5629  df-rel 5630  df-cnv 5631  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-ec 8637  df-qs 8641

This theorem is referenced by:  disjqmap  38997