Theorem eldisjs7 39144

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 39143, but without explicitly mentioning QMap. This is the "E*/E!"" view that is closest in spirit to suc11reg 9532-style injectivity and to the "unique generator per block" narrative. It is also the right contrast-point to older one-line criteria like dfdisjs4 38999 (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 39143	. 2 ⊢ (𝑅 ∈ Disjs ↔ (𝑅 ∈ Rels ∧ (ran QMap 𝑅 ∈ ElDisjs ∧ QMap 𝑅 ∈ Disjs )))
2		qmapex 38654	. . . . . 6 ⊢ (𝑅 ∈ Rels → QMap 𝑅 ∈ V)
3		rnexg 7846	. . . . . 6 ⊢ ( QMap 𝑅 ∈ V → ran QMap 𝑅 ∈ V)
4		eleldisjseldisj 39032	. . . . . 6 ⊢ (ran QMap 𝑅 ∈ V → (ran QMap 𝑅 ∈ ElDisjs ↔ ElDisj ran QMap 𝑅))
5	2, 3, 4	3syl 18	. . . . 5 ⊢ (𝑅 ∈ Rels → (ran QMap 𝑅 ∈ ElDisjs ↔ ElDisj ran QMap 𝑅))
6		rnqmap 38657	. . . . . . 7 ⊢ ran QMap 𝑅 = (dom 𝑅 / 𝑅)
7	6	eldisjeqi 39045	. . . . . 6 ⊢ ( ElDisj ran QMap 𝑅 ↔ ElDisj (dom 𝑅 / 𝑅))
8		dfeldisj4 39015	. . . . . 6 ⊢ ( ElDisj (dom 𝑅 / 𝑅) ↔ ∀𝑥∃*𝑢 ∈ (dom 𝑅 / 𝑅)𝑥 ∈ 𝑢)
9	7, 8	bitri 275	. . . . 5 ⊢ ( ElDisj ran QMap 𝑅 ↔ ∀𝑥∃*𝑢 ∈ (dom 𝑅 / 𝑅)𝑥 ∈ 𝑢)
10	5, 9	bitrdi 287	. . . 4 ⊢ (𝑅 ∈ Rels → (ran QMap 𝑅 ∈ ElDisjs ↔ ∀𝑥∃*𝑢 ∈ (dom 𝑅 / 𝑅)𝑥 ∈ 𝑢))
11		qmapeldisjs 39028	. . . . 5 ⊢ (𝑅 ∈ Rels → ( QMap 𝑅 ∈ Disjs ↔ Disj QMap 𝑅))
12		disjqmap 39030	. . . . 5 ⊢ (𝑅 ∈ Rels → ( Disj QMap 𝑅 ↔ ∀𝑢 ∈ (dom 𝑅 / 𝑅)∃!𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅))
13	11, 12	bitrd 279	. . . 4 ⊢ (𝑅 ∈ Rels → ( QMap 𝑅 ∈ Disjs ↔ ∀𝑢 ∈ (dom 𝑅 / 𝑅)∃!𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅))
14	10, 13	anbi12d 633	. . 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 574	. 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 275	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 206   ∧ wa 395  ∀wal 1540   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃!wreu 3349  ∃*wrmo 3350  Vcvv 3441  dom cdm 5625  ran crn 5626  [cec 8635   / cqs 8636   QMap cqmap 38378   Rels crels 38388   Disjs cdisjs 38421   Disj wdisjALTV 38422   ElDisjs celdisjs 38423   ElDisj weldisj 38424

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 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-eprel 5525  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ec 8639  df-qs 8643  df-rels 38643  df-qmap 38649  df-coss 38704  df-ssr 38781  df-refrel 38795  df-cnvrefs 38808  df-cnvrefrels 38809  df-cnvrefrel 38810  df-symrel 38827  df-trrel 38861  df-eqvrel 38872  df-funALTV 38970  df-disjss 38991  df-disjs 38992  df-disjALTV 38993  df-eldisjs 38994  df-eldisj 38995

This theorem is referenced by:  dfdisjs7  39146