Theorem eldisjsim5 39109

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 39104	. . . . 5 ⊢ (𝑅 ∈ Disjs → Disj 𝑅)
2		disjimrmoeqec 38978	. . . . 5 ⊢ ( Disj 𝑅 → ∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅)
3	1, 2	syl 17	. . . 4 ⊢ (𝑅 ∈ Disjs → ∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅)
4	3	alrimiv 1929	. . 3 ⊢ (𝑅 ∈ Disjs → ∀𝑢∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅)
5		disjqmap2 38996	. . 3 ⊢ (𝑅 ∈ Disjs → ( Disj QMap 𝑅 ↔ ∀𝑢∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅))
6	4, 5	mpbird 257	. 2 ⊢ (𝑅 ∈ Disjs → Disj QMap 𝑅)
7		qmapeldisjs 38995	. 2 ⊢ (𝑅 ∈ Disjs → ( QMap 𝑅 ∈ Disjs ↔ Disj QMap 𝑅))
8	6, 7	mpbird 257	1 ⊢ (𝑅 ∈ Disjs → QMap 𝑅 ∈ Disjs )

Syntax hints:   → wi 4  ∀wal 1540   = wceq 1542   ∈ wcel 2114  ∃*wrmo 3348  dom cdm 5623  [cec 8633   QMap cqmap 38345   Disjs cdisjs 38388   Disj wdisjALTV 38389

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-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-rels 38610  df-qmap 38616  df-coss 38671  df-ssr 38748  df-cnvrefs 38775  df-cnvrefrels 38776  df-cnvrefrel 38777  df-funALTV 38937  df-disjss 38958  df-disjs 38959  df-disjALTV 38960

This theorem is referenced by:  eldisjs6  39110