Theorem disjimrmoeqec 39660

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 39656	. . 3 ⊢ ( Disj 𝑅 → ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 → 𝑢 = 𝑣))
2		eqtr2 2781	. . . . 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 3132	. . 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 8735	. . . 4 ⊢ (𝑢 = 𝑣 → [𝑢]𝑅 = [𝑣]𝑅)
7	6	eqeq2d 2771	. . 3 ⊢ (𝑢 = 𝑣 → (𝑡 = [𝑢]𝑅 ↔ 𝑡 = [𝑣]𝑅))
8	7	rmo4 3687	. 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 401   = wceq 1570  ∀wral 3076  ∃*wrmo 3364  dom cdm 5647  [cec 8693   Disj wdisjALTV 39071

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  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 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-ec 8697  df-coss 39353  df-cnvrefrel 39459  df-disjALTV 39642

This theorem is used by:  disjimdmqseq  39661  eldisjsim5  39791