Theorem disjqmap 39209

Description: Disjointness of QMap equals unique generation of the quotient carrier. The cleaned, carrier-respecting version of disjqmap2 39208. 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 39208	. 2 ⊢ (𝑅 ∈ 𝑉 → ( Disj QMap 𝑅 ↔ ∀𝑢∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅))
2		raldmqseu 38747	. 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 1546   = wceq 1548   ∈ wcel 2121  ∀wral 3055  ∃!wreu 3344  ∃*wrmo 3345  dom cdm 5621  [cec 8635   / cqs 8636   QMap cqmap 38557   Disj wdisjALTV 38601

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-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-fv 6497  df-ec 8639  df-qs 8643  df-qmap 38828  df-coss 38883  df-cnvrefrel 38989  df-funALTV 39149  df-disjALTV 39172

This theorem is referenced by:  eldisjs7  39323