Theorem dfdisjs7 39146

Description: Alternate definition of the class of disjoints (via carrier disjointness + unique representatives). Ideology-free normal form of dfdisjs6 39145: "blocks cover their elements" (∃*) and "each block has a unique generator" (∃!), expressed entirely at the quotient-carrier level. Same class as dfdisjs6 39145, 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 39171 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 39144	. 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 38459	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 3052  ∃!wreu 3349  ∃*wrmo 3350  {crab 3400  dom cdm 5625  [cec 8635   / cqs 8636   Rels crels 38388   Disjs cdisjs 38421

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 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

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 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-eprel 5525  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ec 8639  df-qs 8643  df-rels 38643  df-qmap 38649  df-coss 38704  df-ssr 38781  df-refrel 38795  df-cnvrefs 38808  df-cnvrefrels 38809  df-cnvrefrel 38810  df-symrel 38827  df-trrel 38861  df-eqvrel 38872  df-funALTV 38970  df-disjss 38991  df-disjs 38992  df-disjALTV 38993  df-eldisjs 38994  df-eldisj 38995

This theorem is referenced by: (None)