Theorem eldisjsim5 39280

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 39275	. . . . 5 ⊢ (𝑅 ∈ Disjs → Disj 𝑅)
2		disjimrmoeqec 39149	. . . . 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 39167	. . 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 39166	. 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 3342  dom cdm 5626  [cec 8636   QMap cqmap 38516   Disjs cdisjs 38559   Disj wdisjALTV 38560

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 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-ec 8640  df-rels 38781  df-qmap 38787  df-coss 38842  df-ssr 38919  df-cnvrefs 38946  df-cnvrefrels 38947  df-cnvrefrel 38948  df-funALTV 39108  df-disjss 39129  df-disjs 39130  df-disjALTV 39131

This theorem is referenced by:  eldisjs6  39281