Theorem eldisjsim5 40062

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 40057	. . . . 5 ⊢ (𝑅 ∈ Disjs → Disj 𝑅)
2		disjimrmoeqec 39931	. . . . 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 1962	. . 3 ⊢ (𝑅 ∈ Disjs → ∀𝑢∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅)
5		disjqmap2 39949	. . 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 39948	. 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 1568   = wceq 1570   ∈ wcel 2147  ∃*wrmo 3368  dom cdm 5655  [cec 8740   QMap cqmap 39298   Disjs cdisjs 39341   Disj wdisjALTV 39342

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-ec 8744  df-rels 39563  df-qmap 39569  df-coss 39624  df-ssr 39701  df-cnvrefs 39728  df-cnvrefrels 39729  df-cnvrefrel 39730  df-funALTV 39890  df-disjss 39911  df-disjs 39912  df-disjALTV 39913

This theorem is used by:  eldisjs6  40063