Theorem disjimrmoeqec 39485

Description: Under Disj, every block has a unique generator (∃* form). If 𝑡 is a block in the quotient sense, then there is a uniquely determined 𝑢 in dom 𝑅 such that 𝑡 = [𝑢]𝑅. This is the existence+uniqueness engine behind Disjs and QMap characterizations: it is the "representative theorem" from which the ∃! forms are obtained. (Contributed by Peter Mazsa, 5-Feb-2026.)

Assertion

Ref	Expression
disjimrmoeqec	⊢ ( Disj 𝑅 → ∃*𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅)

Distinct variable groups:   𝑢,𝑅   𝑢,𝑡

Allowed substitution hint:   𝑅(𝑡)

Proof of Theorem disjimrmoeqec

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		disjimeceqim 39481	. . 3 ⊢ ( Disj 𝑅 → ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 → 𝑢 = 𝑣))
2		eqtr2 2783	. . . . 5 ⊢ ((𝑡 = [𝑢]𝑅 ∧ 𝑡 = [𝑣]𝑅) → [𝑢]𝑅 = [𝑣]𝑅)
3	2	imim1i 64	. . . 4 ⊢ (([𝑢]𝑅 = [𝑣]𝑅 → 𝑢 = 𝑣) → ((𝑡 = [𝑢]𝑅 ∧ 𝑡 = [𝑣]𝑅) → 𝑢 = 𝑣))
4	3	2ralimi 3134	. . 3 ⊢ (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 → 𝑢 = 𝑣) → ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅((𝑡 = [𝑢]𝑅 ∧ 𝑡 = [𝑣]𝑅) → 𝑢 = 𝑣))
5	1, 4	syl 18	. 2 ⊢ ( Disj 𝑅 → ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅((𝑡 = [𝑢]𝑅 ∧ 𝑡 = [𝑣]𝑅) → 𝑢 = 𝑣))
6		eceq1 8732	. . . 4 ⊢ (𝑢 = 𝑣 → [𝑢]𝑅 = [𝑣]𝑅)
7	6	eqeq2d 2773	. . 3 ⊢ (𝑢 = 𝑣 → (𝑡 = [𝑢]𝑅 ↔ 𝑡 = [𝑣]𝑅))
8	7	rmo4 3692	. 2 ⊢ (∃*𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅 ↔ ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅((𝑡 = [𝑢]𝑅 ∧ 𝑡 = [𝑣]𝑅) → 𝑢 = 𝑣))
9	5, 8	sylibr 237	1 ⊢ ( Disj 𝑅 → ∃*𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569  ∀wral 3078  ∃*wrmo 3367  dom cdm 5660  [cec 8690   Disj wdisjALTV 38896

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3368  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-ec 8694  df-coss 39178  df-cnvrefrel 39284  df-disjALTV 39467

This theorem is used by:  disjimdmqseq  39486  eldisjsim5  39616