Theorem disjimdmqseq 39933

Description: Disjointness implies unique-generation of quotient blocks. Converts existence-quotient comprehension (see df-qs 8751) 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 39932	. . . 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 8812	. . . . . 6 ⊢ (𝑡 ∈ V → (𝑡 ∈ (dom 𝑅 / 𝑅) ↔ ∃𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅))
4	3	elv 3459	. . . . 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 3371	. . . 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 2897	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 1570   ∈ wcel 2147  {cab 2741  ∃wrex 3090  ∃!wreu 3367  ∃*wrmo 3368  Vcvv 3454  dom cdm 5655  [cec 8743   / cqs 8744   Disj wdisjALTV 39343

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ec 8747  df-qs 8751  df-coss 39625  df-cnvrefrel 39731  df-disjALTV 39914

This theorem is used by: (None)