Theorem dfdisjs6 39263

Description: Alternate definition of the class of disjoints (via quotient-map stability). Disjs is the class of relations 𝑟 whose quotient-map QMap 𝑟 is again disjoint and whose induced quotient-carrier is element-disjoint. This is the definitional "stability-by-decomposition" packaging of disjointness: it builds Disjs from two internal layers (i) a carrier-layer constraint and (ii) a map-layer closure constraint. This is deliberately different from "u R x" style definitions: it makes the carrier of blocks and the uniqueness-of-representatives discipline first-class and reusable (via QMap) rather than implicit. (Contributed by Peter Mazsa, 16-Feb-2026.)

Assertion

Ref	Expression
dfdisjs6	⊢ Disjs = {𝑟 ∈ Rels ∣ (ran QMap 𝑟 ∈ ElDisjs ∧ QMap 𝑟 ∈ Disjs )}

Proof of Theorem dfdisjs6

Step	Hyp	Ref	Expression
1		eldisjs6 39261	. 2 ⊢ (𝑟 ∈ Disjs ↔ (𝑟 ∈ Rels ∧ (ran QMap 𝑟 ∈ ElDisjs ∧ QMap 𝑟 ∈ Disjs )))
2	1	eqrabi 38577	1 ⊢ Disjs = {𝑟 ∈ Rels ∣ (ran QMap 𝑟 ∈ ElDisjs ∧ QMap 𝑟 ∈ Disjs )}

Syntax hints:   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {crab 3389  ran crn 5632   QMap cqmap 38496   Rels crels 38506   Disjs cdisjs 38539   ElDisjs celdisjs 38541

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: (None)