Theorem raldmqsmo 38745

Description: On the quotient carrier, "at most one" and "exactly one" coincide for coset witnesses. (Contributed by Peter Mazsa, 6-Feb-2026.)

Assertion

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

Distinct variable group:   𝑡,𝑅,𝑢

Proof of Theorem raldmqsmo

Step	Hyp	Ref	Expression
1		df-qs 8643	. . . . . 6 ⊢ (dom 𝑅 / 𝑅) = {𝑢 ∣ ∃𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅}
2	1	eqabri 2883	. . . . 5 ⊢ (𝑢 ∈ (dom 𝑅 / 𝑅) ↔ ∃𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅)
3	2	biimpi 218	. . . 4 ⊢ (𝑢 ∈ (dom 𝑅 / 𝑅) → ∃𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅)
4	3	biantrurd 538	. . 3 ⊢ (𝑢 ∈ (dom 𝑅 / 𝑅) → (∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅 ↔ (∃𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅 ∧ ∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅)))
5		reu5 3348	. . 3 ⊢ (∃!𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅 ↔ (∃𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅 ∧ ∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅))
6	4, 5	bitr4di 291	. 2 ⊢ (𝑢 ∈ (dom 𝑅 / 𝑅) → (∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅 ↔ ∃!𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅))
7	6	ralbiia 3085	1 ⊢ (∀𝑢 ∈ (dom 𝑅 / 𝑅)∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅 ↔ ∀𝑢 ∈ (dom 𝑅 / 𝑅)∃!𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅)

Syntax hints:   ↔ wb 208   ∧ wa 397   = wceq 1548   ∈ wcel 2121  ∀wral 3055  ∃wrex 3065  ∃!wreu 3344  ∃*wrmo 3345  dom cdm 5621  [cec 8635   / cqs 8636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-12 2191  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-tru 1551  df-ex 1788  df-sb 2075  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-qs 8643

This theorem is referenced by:  raldmqseu  38747