Theorem eldisjs7 39540

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 39539, but without explicitly mentioning QMap. This is the "E*/E!"" view that is closest in spirit to suc11reg 9591-style injectivity and to the "unique generator per block" narrative. It is also the right contrast-point to older one-line criteria like dfdisjs4 39395 (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 39539	. 2 ⊢ (𝑅 ∈ Disjs ↔ (𝑅 ∈ Rels ∧ (ran QMap 𝑅 ∈ ElDisjs ∧ QMap 𝑅 ∈ Disjs )))
2		qmapex 39050	. . . . . 6 ⊢ (𝑅 ∈ Rels → QMap 𝑅 ∈ V)
3		rnexg 7902	. . . . . 6 ⊢ ( QMap 𝑅 ∈ V → ran QMap 𝑅 ∈ V)
4		eleldisjseldisj 39428	. . . . . 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 39053	. . . . . . 7 ⊢ ran QMap 𝑅 = (dom 𝑅 / 𝑅)
7	6	eldisjeqi 39441	. . . . . 6 ⊢ ( ElDisj ran QMap 𝑅 ↔ ElDisj (dom 𝑅 / 𝑅))
8		dfeldisj4 39411	. . . . . 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 39424	. . . . 5 ⊢ (𝑅 ∈ Rels → ( QMap 𝑅 ∈ Disjs ↔ Disj QMap 𝑅))
12		disjqmap 39426	. . . . 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 643	. . 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 584	. 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)))

Syntax hints:   ↔ wb 209   ∧ wa 400  ∀wal 1566   = wceq 1568   ∈ wcel 2150  ∀wral 3086  ∃!wreu 3374  ∃*wrmo 3375  Vcvv 3462  dom cdm 5665  ran crn 5666  [cec 8695   / cqs 8696   QMap cqmap 38774   Rels crels 38784   Disjs cdisjs 38817   Disj wdisjALTV 38818   ElDisjs celdisjs 38819   ElDisj weldisj 38820

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-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5340  ax-pr 5408  ax-un 7736

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-reu 3377  df-rab 3424  df-v 3464  df-sbc 3753  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5560  df-eprel 5565  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-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ec 8699  df-qs 8703  df-rels 39039  df-qmap 39045  df-coss 39100  df-ssr 39177  df-refrel 39191  df-cnvrefs 39204  df-cnvrefrels 39205  df-cnvrefrel 39206  df-symrel 39223  df-trrel 39257  df-eqvrel 39268  df-funALTV 39366  df-disjss 39387  df-disjs 39388  df-disjALTV 39389  df-eldisjs 39390  df-eldisj 39391

This theorem is referenced by:  dfdisjs7  39542