Theorem eldisjsim5 39616

Description: Disjs is closed under QMap. If a relation is "disjoint-structured" (Disjs), then its canonical block map is also "disjoint-structured". This is the second "structure level" in Disjs: it expresses that the property is stable under passing to the canonical block map, a theme that mirrors Pet-grade stability at a different axis. (Contributed by Peter Mazsa, 15-Feb-2026.)

Assertion

Ref	Expression
eldisjsim5	⊢ (𝑅 ∈ Disjs → QMap 𝑅 ∈ Disjs )

Proof of Theorem eldisjsim5

Dummy variables 𝑡 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldisjsim1 39611	. . . . 5 ⊢ (𝑅 ∈ Disjs → Disj 𝑅)
2		disjimrmoeqec 39485	. . . . 5 ⊢ ( Disj 𝑅 → ∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅)
3	1, 2	syl 18	. . . 4 ⊢ (𝑅 ∈ Disjs → ∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅)
4	3	alrimiv 1956	. . 3 ⊢ (𝑅 ∈ Disjs → ∀𝑢∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅)
5		disjqmap2 39503	. . 3 ⊢ (𝑅 ∈ Disjs → ( Disj QMap 𝑅 ↔ ∀𝑢∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅))
6	4, 5	mpbird 260	. 2 ⊢ (𝑅 ∈ Disjs → Disj QMap 𝑅)
7		qmapeldisjs 39502	. 2 ⊢ (𝑅 ∈ Disjs → ( QMap 𝑅 ∈ Disjs ↔ Disj QMap 𝑅))
8	6, 7	mpbird 260	1 ⊢ (𝑅 ∈ Disjs → QMap 𝑅 ∈ Disjs )

This proof depends on syntax axioms:   → wi 4  ∀wal 1567   = wceq 1569   ∈ wcel 2142  ∃*wrmo 3367  dom cdm 5660  [cec 8690   QMap cqmap 38852   Disjs cdisjs 38895   Disj wdisjALTV 38896

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ec 8694  df-rels 39117  df-qmap 39123  df-coss 39178  df-ssr 39255  df-cnvrefs 39282  df-cnvrefrels 39283  df-cnvrefrel 39284  df-funALTV 39444  df-disjss 39465  df-disjs 39466  df-disjALTV 39467

This theorem is used by:  eldisjs6  39617