Theorem eldisjsim5 39321

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 39316	. . . . 5 ⊢ (𝑅 ∈ Disjs → Disj 𝑅)
2		disjimrmoeqec 39190	. . . . 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 1935	. . 3 ⊢ (𝑅 ∈ Disjs → ∀𝑢∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅)
5		disjqmap2 39208	. . 3 ⊢ (𝑅 ∈ Disjs → ( Disj QMap 𝑅 ↔ ∀𝑢∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅))
6	4, 5	mpbird 259	. 2 ⊢ (𝑅 ∈ Disjs → Disj QMap 𝑅)
7		qmapeldisjs 39207	. 2 ⊢ (𝑅 ∈ Disjs → ( QMap 𝑅 ∈ Disjs ↔ Disj QMap 𝑅))
8	6, 7	mpbird 259	1 ⊢ (𝑅 ∈ Disjs → QMap 𝑅 ∈ Disjs )

Syntax hints:   → wi 4  ∀wal 1546   = wceq 1548   ∈ wcel 2121  ∃*wrmo 3345  dom cdm 5621  [cec 8635   QMap cqmap 38557   Disjs cdisjs 38600   Disj wdisjALTV 38601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ec 8639  df-rels 38822  df-qmap 38828  df-coss 38883  df-ssr 38960  df-cnvrefs 38987  df-cnvrefrels 38988  df-cnvrefrel 38989  df-funALTV 39149  df-disjss 39170  df-disjs 39171  df-disjALTV 39172

This theorem is referenced by:  eldisjs6  39322