Theorem dfdisjs6 39222

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 39220	. 2 ⊢ (𝑟 ∈ Disjs ↔ (𝑟 ∈ Rels ∧ (ran QMap 𝑟 ∈ ElDisjs ∧ QMap 𝑟 ∈ Disjs )))
2	1	eqrabi 38536	1 ⊢ Disjs = {𝑟 ∈ Rels ∣ (ran QMap 𝑟 ∈ ElDisjs ∧ QMap 𝑟 ∈ Disjs )}

Syntax hints:   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {crab 3401  ran crn 5635   QMap cqmap 38455   Rels crels 38465   Disjs cdisjs 38498   ElDisjs celdisjs 38500

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 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692

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 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-eprel 5534  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-ec 8649  df-qs 8653  df-rels 38720  df-qmap 38726  df-coss 38781  df-ssr 38858  df-refrel 38872  df-cnvrefs 38885  df-cnvrefrels 38886  df-cnvrefrel 38887  df-symrel 38904  df-trrel 38938  df-eqvrel 38949  df-funALTV 39047  df-disjss 39068  df-disjs 39069  df-disjALTV 39070  df-eldisjs 39071  df-eldisj 39072

This theorem is referenced by: (None)