Theorem disjimdmqseq 39191

Description: Disjointness implies unique-generation of quotient blocks. Converts existence-quotient comprehension (see df-qs 8643) 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 39190	. . . 4 ⊢ ( Disj 𝑅 → ∃*𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅)
2	1	biantrud 537	. . 3 ⊢ ( Disj 𝑅 → (𝑡 ∈ (dom 𝑅 / 𝑅) ↔ (𝑡 ∈ (dom 𝑅 / 𝑅) ∧ ∃*𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅)))
3		elqsg 8704	. . . . . 6 ⊢ (𝑡 ∈ V → (𝑡 ∈ (dom 𝑅 / 𝑅) ↔ ∃𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅))
4	3	elv 3438	. . . . 5 ⊢ (𝑡 ∈ (dom 𝑅 / 𝑅) ↔ ∃𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅)
5	4	anbi1i 631	. . . 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 3348	. . . 4 ⊢ (∃!𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅 ↔ (∃𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅 ∧ ∃*𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅))
7	5, 6	bitr4i 280	. . 3 ⊢ ((𝑡 ∈ (dom 𝑅 / 𝑅) ∧ ∃*𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅) ↔ ∃!𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅)
8	2, 7	bitrdi 289	. 2 ⊢ ( Disj 𝑅 → (𝑡 ∈ (dom 𝑅 / 𝑅) ↔ ∃!𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅))
9	8	eqabdv 2874	1 ⊢ ( Disj 𝑅 → (dom 𝑅 / 𝑅) = {𝑡 ∣ ∃!𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   = wceq 1548   ∈ wcel 2121  {cab 2719  ∃wrex 3065  ∃!wreu 3344  ∃*wrmo 3345  Vcvv 3433  dom cdm 5621  [cec 8635   / cqs 8636   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-pr 5365

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-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-br 5076  df-opab 5138  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-ec 8639  df-qs 8643  df-coss 38883  df-cnvrefrel 38989  df-disjALTV 39172

This theorem is referenced by: (None)