Theorem disjqmap 39331

Description: Disjointness of QMap equals unique generation of the quotient carrier. The cleaned, carrier-respecting version of disjqmap2 39330. This is the statement "each equivalence class has a unique representative" for the general coset carrier (dom 𝑅 / 𝑅). (Contributed by Peter Mazsa, 12-Feb-2026.)

Assertion

Ref	Expression
disjqmap	⊢ (𝑅 ∈ 𝑉 → ( Disj QMap 𝑅 ↔ ∀𝑢 ∈ (dom 𝑅 / 𝑅)∃!𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅))

Distinct variable groups:   𝑡,𝑅,𝑢   𝑡,𝑉,𝑢

Proof of Theorem disjqmap

Step	Hyp	Ref	Expression
1		disjqmap2 39330	. 2 ⊢ (𝑅 ∈ 𝑉 → ( Disj QMap 𝑅 ↔ ∀𝑢∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅))
2		raldmqseu 38869	. 2 ⊢ (𝑅 ∈ 𝑉 → (∀𝑢 ∈ (dom 𝑅 / 𝑅)∃!𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅 ↔ ∀𝑢∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅))
3	1, 2	bitr4d 284	1 ⊢ (𝑅 ∈ 𝑉 → ( Disj QMap 𝑅 ↔ ∀𝑢 ∈ (dom 𝑅 / 𝑅)∃!𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1560   = wceq 1562   ∈ wcel 2144  ∀wral 3078  ∃!wreu 3367  ∃*wrmo 3368  dom cdm 5649  [cec 8678   / cqs 8679   QMap cqmap 38679   Disj wdisjALTV 38723

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-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  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-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  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-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-fv 6531  df-ec 8682  df-qs 8686  df-qmap 38950  df-coss 39005  df-cnvrefrel 39111  df-funALTV 39271  df-disjALTV 39294

This theorem is referenced by:  eldisjs7  39445