Theorem disjimdmqseq 39817

Description: Disjointness implies unique-generation of quotient blocks. Converts existence-quotient comprehension (see df-qs 8724) 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 39816	. . . 4 ⊢ ( Disj 𝑅 → ∃*𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅)
2	1	biantrud 541	. . 3 ⊢ ( Disj 𝑅 → (𝑡 ∈ (dom 𝑅 / 𝑅) ↔ (𝑡 ∈ (dom 𝑅 / 𝑅) ∧ ∃*𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅)))
3		elqsg 8785	. . . . . 6 ⊢ (𝑡 ∈ V → (𝑡 ∈ (dom 𝑅 / 𝑅) ↔ ∃𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅))
4	3	elv 3455	. . . . 5 ⊢ (𝑡 ∈ (dom 𝑅 / 𝑅) ↔ ∃𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅)
5	4	anbi1i 636	. . . 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 3367	. . . 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 2893	1 ⊢ ( Disj 𝑅 → (dom 𝑅 / 𝑅) = {𝑡 ∣ ∃!𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {cab 2738  ∃wrex 3086  ∃!wreu 3363  ∃*wrmo 3364  Vcvv 3450  dom cdm 5650  [cec 8716   / cqs 8717   Disj wdisjALTV 39227

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ec 8720  df-qs 8724  df-coss 39509  df-cnvrefrel 39615  df-disjALTV 39798

This theorem is used by: (None)