Theorem dfdisjs7 39113

Description: Alternate definition of the class of disjoints (via carrier disjointness + unique representatives). Ideology-free normal form of dfdisjs6 39112: "blocks cover their elements" (∃*) and "each block has a unique generator" (∃!), expressed entirely at the quotient-carrier level. Same class as dfdisjs6 39112, but presented in fully expanded ∃* / ∃! form over the quotient-carrier (dom 𝑟 / 𝑟). Makes explicit (a) element-disjointness of the quotient-carrier and (b) unique representative existence for each block. These are exactly the two conditions that rule out type-confusions (blocks vs witnesses) and ensure canonical decomposition. This is the form that best supports analogy arguments with df-petparts 39138 and with successor-style uniqueness patterns. (Contributed by Peter Mazsa, 16-Feb-2026.)

Assertion

Ref	Expression
dfdisjs7	⊢ 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 dfdisjs7

Step	Hyp	Ref	Expression
1		eldisjs7 39111	. 2 ⊢ (𝑟 ∈ Disjs ↔ (𝑟 ∈ Rels ∧ (∀𝑥∃*𝑢 ∈ (dom 𝑟 / 𝑟)𝑥 ∈ 𝑢 ∧ ∀𝑢 ∈ (dom 𝑟 / 𝑟)∃!𝑡 ∈ dom 𝑟 𝑢 = [𝑡]𝑟)))
2	1	eqrabi 38427	1 ⊢ Disjs = {𝑟 ∈ Rels ∣ (∀𝑥∃*𝑢 ∈ (dom 𝑟 / 𝑟)𝑥 ∈ 𝑢 ∧ ∀𝑢 ∈ (dom 𝑟 / 𝑟)∃!𝑡 ∈ dom 𝑟 𝑢 = [𝑡]𝑟)}

Syntax hints:   ∧ wa 395  ∀wal 1540   = wceq 1542   ∈ wcel 2114  ∀wral 3050  ∃!wreu 3347  ∃*wrmo 3348  {crab 3398  dom cdm 5623  [cec 8633   / cqs 8634   Rels crels 38355   Disjs cdisjs 38388

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 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680

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 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5518  df-eprel 5523  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-ec 8637  df-qs 8641  df-rels 38610  df-qmap 38616  df-coss 38671  df-ssr 38748  df-refrel 38762  df-cnvrefs 38775  df-cnvrefrels 38776  df-cnvrefrel 38777  df-symrel 38794  df-trrel 38828  df-eqvrel 38839  df-funALTV 38937  df-disjss 38958  df-disjs 38959  df-disjALTV 38960  df-eldisjs 38961  df-eldisj 38962

This theorem is referenced by: (None)