Theorem disjimrmoeqec 39407

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 39403	. . 3 ⊢ ( Disj 𝑅 → ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 → 𝑢 = 𝑣))
2		eqtr2 2791	. . . . 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 3142	. . 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 8737	. . . 4 ⊢ (𝑢 = 𝑣 → [𝑢]𝑅 = [𝑣]𝑅)
7	6	eqeq2d 2781	. . 3 ⊢ (𝑢 = 𝑣 → (𝑡 = [𝑢]𝑅 ↔ 𝑡 = [𝑣]𝑅))
8	7	rmo4 3701	. 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)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1568  ∀wral 3086  ∃*wrmo 3375  dom cdm 5665  [cec 8695   Disj wdisjALTV 38818

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-sep 5262  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-ral 3087  df-rex 3097  df-rmo 3376  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-id 5560  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-ec 8699  df-coss 39100  df-cnvrefrel 39206  df-disjALTV 39389

This theorem is referenced by:  disjimdmqseq  39408  eldisjsim5  39538