Theorem raldmqsmo 38533

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 8641	. . . . . 6 ⊢ (dom 𝑅 / 𝑅) = {𝑢 ∣ ∃𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅}
2	1	eqabri 2877	. . . . 5 ⊢ (𝑢 ∈ (dom 𝑅 / 𝑅) ↔ ∃𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅)
3	2	biimpi 216	. . . 4 ⊢ (𝑢 ∈ (dom 𝑅 / 𝑅) → ∃𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅)
4	3	biantrurd 532	. . 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 3351	. . 3 ⊢ (∃!𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅 ↔ (∃𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅 ∧ ∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅))
6	4, 5	bitr4di 289	. 2 ⊢ (𝑢 ∈ (dom 𝑅 / 𝑅) → (∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅 ↔ ∃!𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅))
7	6	ralbiia 3079	1 ⊢ (∀𝑢 ∈ (dom 𝑅 / 𝑅)∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅 ↔ ∀𝑢 ∈ (dom 𝑅 / 𝑅)∃!𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3050  ∃wrex 3059  ∃!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

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-qs 8641

This theorem is referenced by:  raldmqseu  38535