Theorem eldisjs7 39262

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 39261, but without explicitly mentioning QMap. This is the "E*/E!"" view that is closest in spirit to suc11reg 9540-style injectivity and to the "unique generator per block" narrative. It is also the right contrast-point to older one-line criteria like dfdisjs4 39117 (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 39261	. 2 ⊢ (𝑅 ∈ Disjs ↔ (𝑅 ∈ Rels ∧ (ran QMap 𝑅 ∈ ElDisjs ∧ QMap 𝑅 ∈ Disjs )))
2		qmapex 38772	. . . . . 6 ⊢ (𝑅 ∈ Rels → QMap 𝑅 ∈ V)
3		rnexg 7853	. . . . . 6 ⊢ ( QMap 𝑅 ∈ V → ran QMap 𝑅 ∈ V)
4		eleldisjseldisj 39150	. . . . . 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 38775	. . . . . . 7 ⊢ ran QMap 𝑅 = (dom 𝑅 / 𝑅)
7	6	eldisjeqi 39163	. . . . . 6 ⊢ ( ElDisj ran QMap 𝑅 ↔ ElDisj (dom 𝑅 / 𝑅))
8		dfeldisj4 39133	. . . . . 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 39146	. . . . 5 ⊢ (𝑅 ∈ Rels → ( QMap 𝑅 ∈ Disjs ↔ Disj QMap 𝑅))
12		disjqmap 39148	. . . . 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 3051  ∃!wreu 3340  ∃*wrmo 3341  Vcvv 3429  dom cdm 5631  ran crn 5632  [cec 8641   / cqs 8642   QMap cqmap 38496   Rels crels 38506   Disjs cdisjs 38539   Disj wdisjALTV 38540   ElDisjs celdisjs 38541   ElDisj weldisj 38542

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-eprel 5531  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ec 8645  df-qs 8649  df-rels 38761  df-qmap 38767  df-coss 38822  df-ssr 38899  df-refrel 38913  df-cnvrefs 38926  df-cnvrefrels 38927  df-cnvrefrel 38928  df-symrel 38945  df-trrel 38979  df-eqvrel 38990  df-funALTV 39088  df-disjss 39109  df-disjs 39110  df-disjALTV 39111  df-eldisjs 39112  df-eldisj 39113

This theorem is referenced by:  dfdisjs7  39264