Theorem disjimdmqseq 39012

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 39011	. . . 4 ⊢ ( Disj 𝑅 → ∃*𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅)
2	1	biantrud 531	. . 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 3446	. . . . 5 ⊢ (𝑡 ∈ (dom 𝑅 / 𝑅) ↔ ∃𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅)
5	4	anbi1i 625	. . . 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 3353	. . . 4 ⊢ (∃!𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅 ↔ (∃𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅 ∧ ∃*𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅))
7	5, 6	bitr4i 278	. . 3 ⊢ ((𝑡 ∈ (dom 𝑅 / 𝑅) ∧ ∃*𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅) ↔ ∃!𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅)
8	2, 7	bitrdi 287	. 2 ⊢ ( Disj 𝑅 → (𝑡 ∈ (dom 𝑅 / 𝑅) ↔ ∃!𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅))
9	8	eqabdv 2870	1 ⊢ ( Disj 𝑅 → (dom 𝑅 / 𝑅) = {𝑡 ∣ ∃!𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715  ∃wrex 3061  ∃!wreu 3349  ∃*wrmo 3350  Vcvv 3441  dom cdm 5625  [cec 8635   / cqs 8636   Disj wdisjALTV 38422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ec 8639  df-qs 8643  df-coss 38704  df-cnvrefrel 38810  df-disjALTV 38993

This theorem is referenced by: (None)