Theorem raldmqsmo 39098

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 8705	. . . . . 6 ⊢ (dom 𝑅 / 𝑅) = {𝑢 ∣ ∃𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅}
2	1	eqabri 2904	. . . . 5 ⊢ (𝑢 ∈ (dom 𝑅 / 𝑅) ↔ ∃𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅)
3	2	biimpi 219	. . . 4 ⊢ (𝑢 ∈ (dom 𝑅 / 𝑅) → ∃𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅)
4	3	biantrurd 542	. . 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 3369	. . 3 ⊢ (∃!𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅 ↔ (∃𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅 ∧ ∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅))
6	4, 5	bitr4di 292	. 2 ⊢ (𝑢 ∈ (dom 𝑅 / 𝑅) → (∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅 ↔ ∃!𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅))
7	6	ralbiia 3108	1 ⊢ (∀𝑢 ∈ (dom 𝑅 / 𝑅)∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅 ↔ ∀𝑢 ∈ (dom 𝑅 / 𝑅)∃!𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ∃wrex 3088  ∃!wreu 3365  ∃*wrmo 3366  dom cdm 5659  [cec 8697   / cqs 8698

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-12 2215  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-qs 8705

This theorem is used by:  raldmqseu  39100