Theorem disjimrmoeqec 38978

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 38974	. . 3 ⊢ ( Disj 𝑅 → ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 → 𝑢 = 𝑣))
2		eqtr2 2756	. . . . 5 ⊢ ((𝑡 = [𝑢]𝑅 ∧ 𝑡 = [𝑣]𝑅) → [𝑢]𝑅 = [𝑣]𝑅)
3	2	imim1i 63	. . . 4 ⊢ (([𝑢]𝑅 = [𝑣]𝑅 → 𝑢 = 𝑣) → ((𝑡 = [𝑢]𝑅 ∧ 𝑡 = [𝑣]𝑅) → 𝑢 = 𝑣))
4	3	2ralimi 3105	. . 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 17	. 2 ⊢ ( Disj 𝑅 → ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅((𝑡 = [𝑢]𝑅 ∧ 𝑡 = [𝑣]𝑅) → 𝑢 = 𝑣))
6		eceq1 8675	. . . 4 ⊢ (𝑢 = 𝑣 → [𝑢]𝑅 = [𝑣]𝑅)
7	6	eqeq2d 2746	. . 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 234	1 ⊢ ( Disj 𝑅 → ∃*𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  ∀wral 3050  ∃*wrmo 3348  dom cdm 5623  [cec 8633   Disj wdisjALTV 38389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3349  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-ec 8637  df-coss 38671  df-cnvrefrel 38777  df-disjALTV 38960

This theorem is referenced by:  disjimdmqseq  38979  eldisjsim5  39109