Theorem eldisjs7 39793

Description: Elementhood in the class of disjoints. 𝑅 ∈ Disjs iff:

𝑅 ∈ Rels, and

every 𝑥 belongs to at most one block 𝑢 in the quotient-carrier (dom 𝑅 / 𝑅) (element-disjointness at the carrier), and

every block 𝑢 in the quotient-carrier has a unique representative 𝑡 ∈ dom 𝑅 such that 𝑢 = [𝑡]𝑅.

Provides the "fully expanded" quantifier characterization of the same decomposition as eldisjs6 39792, but without explicitly mentioning QMap. This is the "E*/E!"" view that is closest in spirit to suc11reg 9598-style injectivity and to the "unique generator per block" narrative. It is also the right contrast-point to older one-line criteria like dfdisjs4 39648 (the "u R x" style), because it makes the carrier and representation discipline explicit and type-safe. (Contributed by Peter Mazsa, 16-Feb-2026.)

Assertion

Ref	Expression
eldisjs7	⊢ (𝑅 ∈ Disjs ↔ (𝑅 ∈ Rels ∧ (∀𝑥∃*𝑢 ∈ (dom 𝑅 / 𝑅)𝑥 ∈ 𝑢 ∧ ∀𝑢 ∈ (dom 𝑅 / 𝑅)∃!𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅)))

Distinct variable groups:   𝑡,𝑅,𝑢   𝑥,𝑅,𝑢

Proof of Theorem eldisjs7

Step	Hyp	Ref	Expression
1		eldisjs6 39792	. 2 ⊢ (𝑅 ∈ Disjs ↔ (𝑅 ∈ Rels ∧ (ran QMap 𝑅 ∈ ElDisjs ∧ QMap 𝑅 ∈ Disjs )))
2		qmapex 39303	. . . . . 6 ⊢ (𝑅 ∈ Rels → QMap 𝑅 ∈ V)
3		rnexg 7897	. . . . . 6 ⊢ ( QMap 𝑅 ∈ V → ran QMap 𝑅 ∈ V)
4		eleldisjseldisj 39681	. . . . . 6 ⊢ (ran QMap 𝑅 ∈ V → (ran QMap 𝑅 ∈ ElDisjs ↔ ElDisj ran QMap 𝑅))
5	2, 3, 4	3syl 19	. . . . 5 ⊢ (𝑅 ∈ Rels → (ran QMap 𝑅 ∈ ElDisjs ↔ ElDisj ran QMap 𝑅))
6		rnqmap 39306	. . . . . . 7 ⊢ ran QMap 𝑅 = (dom 𝑅 / 𝑅)
7	6	eldisjeqi 39694	. . . . . 6 ⊢ ( ElDisj ran QMap 𝑅 ↔ ElDisj (dom 𝑅 / 𝑅))
8		dfeldisj4 39664	. . . . . 6 ⊢ ( ElDisj (dom 𝑅 / 𝑅) ↔ ∀𝑥∃*𝑢 ∈ (dom 𝑅 / 𝑅)𝑥 ∈ 𝑢)
9	7, 8	bitri 278	. . . . 5 ⊢ ( ElDisj ran QMap 𝑅 ↔ ∀𝑥∃*𝑢 ∈ (dom 𝑅 / 𝑅)𝑥 ∈ 𝑢)
10	5, 9	bitrdi 290	. . . 4 ⊢ (𝑅 ∈ Rels → (ran QMap 𝑅 ∈ ElDisjs ↔ ∀𝑥∃*𝑢 ∈ (dom 𝑅 / 𝑅)𝑥 ∈ 𝑢))
11		qmapeldisjs 39677	. . . . 5 ⊢ (𝑅 ∈ Rels → ( QMap 𝑅 ∈ Disjs ↔ Disj QMap 𝑅))
12		disjqmap 39679	. . . . 5 ⊢ (𝑅 ∈ Rels → ( Disj QMap 𝑅 ↔ ∀𝑢 ∈ (dom 𝑅 / 𝑅)∃!𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅))
13	11, 12	bitrd 282	. . . 4 ⊢ (𝑅 ∈ Rels → ( QMap 𝑅 ∈ Disjs ↔ ∀𝑢 ∈ (dom 𝑅 / 𝑅)∃!𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅))
14	10, 13	anbi12d 644	. . 3 ⊢ (𝑅 ∈ Rels → ((ran QMap 𝑅 ∈ ElDisjs ∧ QMap 𝑅 ∈ Disjs ) ↔ (∀𝑥∃*𝑢 ∈ (dom 𝑅 / 𝑅)𝑥 ∈ 𝑢 ∧ ∀𝑢 ∈ (dom 𝑅 / 𝑅)∃!𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅)))
15	14	pm5.32i 585	. 2 ⊢ ((𝑅 ∈ Rels ∧ (ran QMap 𝑅 ∈ ElDisjs ∧ QMap 𝑅 ∈ Disjs )) ↔ (𝑅 ∈ Rels ∧ (∀𝑥∃*𝑢 ∈ (dom 𝑅 / 𝑅)𝑥 ∈ 𝑢 ∧ ∀𝑢 ∈ (dom 𝑅 / 𝑅)∃!𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅)))
16	1, 15	bitri 278	1 ⊢ (𝑅 ∈ Disjs ↔ (𝑅 ∈ Rels ∧ (∀𝑥∃*𝑢 ∈ (dom 𝑅 / 𝑅)𝑥 ∈ 𝑢 ∧ ∀𝑢 ∈ (dom 𝑅 / 𝑅)∃!𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃!wreu 3363  ∃*wrmo 3364  Vcvv 3450  dom cdm 5647  ran crn 5648  [cec 8693   / cqs 8694   QMap cqmap 39027   Rels crels 39037   Disjs cdisjs 39070   Disj wdisjALTV 39071   ElDisjs celdisjs 39072   ElDisj weldisj 39073

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-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734

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-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-eprel 5547  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-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ec 8697  df-qs 8701  df-rels 39292  df-qmap 39298  df-coss 39353  df-ssr 39430  df-refrel 39444  df-cnvrefs 39457  df-cnvrefrels 39458  df-cnvrefrel 39459  df-symrel 39476  df-trrel 39510  df-eqvrel 39521  df-funALTV 39619  df-disjss 39640  df-disjs 39641  df-disjALTV 39642  df-eldisjs 39643  df-eldisj 39644

This theorem is used by:  dfdisjs7  39795