Theorem raldmqseu 38869

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 38867	. . 3 ⊢ (∀𝑢 ∈ (dom 𝑅 / 𝑅)∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅 ↔ ∀𝑢 ∈ (dom 𝑅 / 𝑅)∃!𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅)
2		ralrmo3 38868	. . 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 279	. 2 ⊢ (∀𝑢 ∈ (dom 𝑅 / 𝑅)∃!𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅 ↔ ∀𝑢∃*𝑡 ∈ dom 𝑅(𝑢 ∈ (dom 𝑅 / 𝑅) ∧ 𝑢 = [𝑡]𝑅))
4		eqelb 38745	. . . . . . 7 ⊢ ((𝑢 = [𝑡]𝑅 ∧ 𝑢 ∈ (dom 𝑅 / 𝑅)) ↔ (𝑢 = [𝑡]𝑅 ∧ [𝑡]𝑅 ∈ (dom 𝑅 / 𝑅)))
5		ancom 464	. . . . . . 7 ⊢ ((𝑢 = [𝑡]𝑅 ∧ 𝑢 ∈ (dom 𝑅 / 𝑅)) ↔ (𝑢 ∈ (dom 𝑅 / 𝑅) ∧ 𝑢 = [𝑡]𝑅))
6		ancom 464	. . . . . . 7 ⊢ ((𝑢 = [𝑡]𝑅 ∧ [𝑡]𝑅 ∈ (dom 𝑅 / 𝑅)) ↔ ([𝑡]𝑅 ∈ (dom 𝑅 / 𝑅) ∧ 𝑢 = [𝑡]𝑅))
7	4, 5, 6	3bitr3i 303	. . . . . 6 ⊢ ((𝑢 ∈ (dom 𝑅 / 𝑅) ∧ 𝑢 = [𝑡]𝑅) ↔ ([𝑡]𝑅 ∈ (dom 𝑅 / 𝑅) ∧ 𝑢 = [𝑡]𝑅))
8		eceldmqs 8771	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → ([𝑡]𝑅 ∈ (dom 𝑅 / 𝑅) ↔ 𝑡 ∈ dom 𝑅))
9	8	anbi1d 640	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (([𝑡]𝑅 ∈ (dom 𝑅 / 𝑅) ∧ 𝑢 = [𝑡]𝑅) ↔ (𝑡 ∈ dom 𝑅 ∧ 𝑢 = [𝑡]𝑅)))
10	7, 9	bitrid 285	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → ((𝑢 ∈ (dom 𝑅 / 𝑅) ∧ 𝑢 = [𝑡]𝑅) ↔ (𝑡 ∈ dom 𝑅 ∧ 𝑢 = [𝑡]𝑅)))
11	10	rmobidv 3384	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (∃*𝑡 ∈ dom 𝑅(𝑢 ∈ (dom 𝑅 / 𝑅) ∧ 𝑢 = [𝑡]𝑅) ↔ ∃*𝑡 ∈ dom 𝑅(𝑡 ∈ dom 𝑅 ∧ 𝑢 = [𝑡]𝑅)))
12		rmoanid 3379	. . . 4 ⊢ (∃*𝑡 ∈ dom 𝑅(𝑡 ∈ dom 𝑅 ∧ 𝑢 = [𝑡]𝑅) ↔ ∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅)
13	11, 12	bitrdi 289	. . 3 ⊢ (𝑅 ∈ 𝑉 → (∃*𝑡 ∈ dom 𝑅(𝑢 ∈ (dom 𝑅 / 𝑅) ∧ 𝑢 = [𝑡]𝑅) ↔ ∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅))
14	13	albidv 1942	. 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 285	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 208   ∧ wa 399  ∀wal 1560   = wceq 1562   ∈ wcel 2144  ∀wral 3078  ∃!wreu 3367  ∃*wrmo 3368  dom cdm 5649  [cec 8678   / cqs 8679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-ne 2960  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-xp 5655  df-rel 5656  df-cnv 5657  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-ec 8682  df-qs 8686

This theorem is referenced by:  disjqmap  39331