Theorem dfdisjs7 39447

Description: Alternate definition of the class of disjoints (via carrier disjointness + unique representatives). Ideology-free normal form of dfdisjs6 39446: "blocks cover their elements" (∃*) and "each block has a unique generator" (∃!), expressed entirely at the quotient-carrier level. Same class as dfdisjs6 39446, 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 39472 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 39445	. 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 38760	1 ⊢ Disjs = {𝑟 ∈ Rels ∣ (∀𝑥∃*𝑢 ∈ (dom 𝑟 / 𝑟)𝑥 ∈ 𝑢 ∧ ∀𝑢 ∈ (dom 𝑟 / 𝑟)∃!𝑡 ∈ dom 𝑟 𝑢 = [𝑡]𝑟)}

Syntax hints:   ∧ wa 399  ∀wal 1560   = wceq 1562   ∈ wcel 2144  ∀wral 3078  ∃!wreu 3367  ∃*wrmo 3368  {crab 3416  dom cdm 5649  [cec 8678   / cqs 8679   Rels crels 38689   Disjs cdisjs 38722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-eprel 5549  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-ec 8682  df-qs 8686  df-rels 38944  df-qmap 38950  df-coss 39005  df-ssr 39082  df-refrel 39096  df-cnvrefs 39109  df-cnvrefrels 39110  df-cnvrefrel 39111  df-symrel 39128  df-trrel 39162  df-eqvrel 39173  df-funALTV 39271  df-disjss 39292  df-disjs 39293  df-disjALTV 39294  df-eldisjs 39295  df-eldisj 39296

This theorem is referenced by: (None)