Theorem raldmqsmo 38962

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 8703	. . . . . 6 ⊢ (dom 𝑅 / 𝑅) = {𝑢 ∣ ∃𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅}
2	1	eqabri 2912	. . . . 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 541	. . 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 3378	. . 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 3116	1 ⊢ (∀𝑢 ∈ (dom 𝑅 / 𝑅)∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅 ↔ ∀𝑢 ∈ (dom 𝑅 / 𝑅)∃!𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅)

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1568   ∈ wcel 2150  ∀wral 3086  ∃wrex 3096  ∃!wreu 3374  ∃*wrmo 3375  dom cdm 5665  [cec 8695   / cqs 8696

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-12 2220  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1571  df-ex 1808  df-sb 2099  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-ral 3087  df-rex 3097  df-rmo 3376  df-reu 3377  df-qs 8703

This theorem is referenced by:  raldmqseu  38964