Theorem disjimdmqseq 39408

Description: Disjointness implies unique-generation of quotient blocks. Converts existence-quotient comprehension (see df-qs 8703) into a uniqueness-comprehension under disjointness; rewrites (dom 𝑅 / 𝑅) carriers as exactly the class of blocks with a unique representative. This is the "unique generator per block" content in a carrier-normal form. (Contributed by Peter Mazsa, 5-Feb-2026.)

Assertion

Ref	Expression
disjimdmqseq	⊢ ( Disj 𝑅 → (dom 𝑅 / 𝑅) = {𝑡 ∣ ∃!𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅})

Distinct variable group:   𝑡,𝑅,𝑢

Proof of Theorem disjimdmqseq

Step	Hyp	Ref	Expression
1		disjimrmoeqec 39407	. . . 4 ⊢ ( Disj 𝑅 → ∃*𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅)
2	1	biantrud 540	. . 3 ⊢ ( Disj 𝑅 → (𝑡 ∈ (dom 𝑅 / 𝑅) ↔ (𝑡 ∈ (dom 𝑅 / 𝑅) ∧ ∃*𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅)))
3		elqsg 8764	. . . . . 6 ⊢ (𝑡 ∈ V → (𝑡 ∈ (dom 𝑅 / 𝑅) ↔ ∃𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅))
4	3	elv 3467	. . . . 5 ⊢ (𝑡 ∈ (dom 𝑅 / 𝑅) ↔ ∃𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅)
5	4	anbi1i 635	. . . 4 ⊢ ((𝑡 ∈ (dom 𝑅 / 𝑅) ∧ ∃*𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅) ↔ (∃𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅 ∧ ∃*𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅))
6		reu5 3378	. . . 4 ⊢ (∃!𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅 ↔ (∃𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅 ∧ ∃*𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅))
7	5, 6	bitr4i 281	. . 3 ⊢ ((𝑡 ∈ (dom 𝑅 / 𝑅) ∧ ∃*𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅) ↔ ∃!𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅)
8	2, 7	bitrdi 290	. 2 ⊢ ( Disj 𝑅 → (𝑡 ∈ (dom 𝑅 / 𝑅) ↔ ∃!𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅))
9	8	eqabdv 2903	1 ⊢ ( Disj 𝑅 → (dom 𝑅 / 𝑅) = {𝑡 ∣ ∃!𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1568   ∈ wcel 2150  {cab 2748  ∃wrex 3096  ∃!wreu 3374  ∃*wrmo 3375  Vcvv 3462  dom cdm 5665  [cec 8695   / cqs 8696   Disj wdisjALTV 38818

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-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-sep 5262  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-ral 3087  df-rex 3097  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-id 5560  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-ec 8699  df-qs 8703  df-coss 39100  df-cnvrefrel 39206  df-disjALTV 39389

This theorem is referenced by: (None)