Theorem dfdisjs6 39794

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 39792	. 2 ⊢ (𝑟 ∈ Disjs ↔ (𝑟 ∈ Rels ∧ (ran QMap 𝑟 ∈ ElDisjs ∧ QMap 𝑟 ∈ Disjs )))
2	1	eqrabi 39108	1 ⊢ Disjs = {𝑟 ∈ Rels ∣ (ran QMap 𝑟 ∈ ElDisjs ∧ QMap 𝑟 ∈ Disjs )}

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {crab 3412  ran crn 5648   QMap cqmap 39027   Rels crels 39037   Disjs cdisjs 39070   ElDisjs celdisjs 39072

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-eprel 5547  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ec 8697  df-qs 8701  df-rels 39292  df-qmap 39298  df-coss 39353  df-ssr 39430  df-refrel 39444  df-cnvrefs 39457  df-cnvrefrels 39458  df-cnvrefrel 39459  df-symrel 39476  df-trrel 39510  df-eqvrel 39521  df-funALTV 39619  df-disjss 39640  df-disjs 39641  df-disjALTV 39642  df-eldisjs 39643  df-eldisj 39644

This theorem is used by: (None)