Theorem blockadjliftmap 38996

Description: A "two-stage" construction is obtained by first forming the block relation (𝑅 ⋉ ^◡ E ) and then adjoining elements as "BlockAdj". Combined, it uses the relation ((𝑅 ⋉ ^◡ E ) ∪ ^◡ E ). (Contributed by Peter Mazsa, 28-Jan-2026.)

Assertion

Ref	Expression
blockadjliftmap	⊢ ((𝑅 ⋉ ◡ E ) AdjLiftMap 𝐴) = {⟨𝑚, 𝑛⟩ ∣ (𝑚 ∈ (𝐴 ∖ {∅}) ∧ 𝑛 = (𝑚 ∪ ([𝑚]𝑅 × 𝑚)))}

Distinct variable groups:   𝐴,𝑚,𝑛   𝑅,𝑚,𝑛

Proof of Theorem blockadjliftmap

Step	Hyp	Ref	Expression
1		dfadjliftmap 38994	. 2 ⊢ ((𝑅 ⋉ ◡ E ) AdjLiftMap 𝐴) = (𝑚 ∈ dom (((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴) ↦ [𝑚](((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴))
2		df-mpt 5197	. 2 ⊢ (𝑚 ∈ dom (((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴) ↦ [𝑚](((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴)) = {⟨𝑚, 𝑛⟩ ∣ (𝑚 ∈ dom (((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴) ∧ 𝑛 = [𝑚](((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴))}
3		dmxrnuncnvepres 38930	. . . . . 6 ⊢ dom (((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴) = (𝐴 ∖ {∅})
4	3	eleq2i 2861	. . . . 5 ⊢ (𝑚 ∈ dom (((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴) ↔ 𝑚 ∈ (𝐴 ∖ {∅}))
5	4	anbi1i 635	. . . 4 ⊢ ((𝑚 ∈ dom (((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴) ∧ 𝑛 = [𝑚](((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴)) ↔ (𝑚 ∈ (𝐴 ∖ {∅}) ∧ 𝑛 = [𝑚](((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴)))
6		eldifi 4093	. . . . . . . 8 ⊢ (𝑚 ∈ (𝐴 ∖ {∅}) → 𝑚 ∈ 𝐴)
7		ecuncnvepres 38933	. . . . . . . 8 ⊢ (𝑚 ∈ 𝐴 → [𝑚](((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴) = (𝑚 ∪ [𝑚](𝑅 ⋉ ◡ E )))
8	6, 7	syl 18	. . . . . . 7 ⊢ (𝑚 ∈ (𝐴 ∖ {∅}) → [𝑚](((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴) = (𝑚 ∪ [𝑚](𝑅 ⋉ ◡ E )))
9		ecxrncnvep2 38948	. . . . . . . . 9 ⊢ (𝑚 ∈ V → [𝑚](𝑅 ⋉ ◡ E ) = ([𝑚]𝑅 × 𝑚))
10	9	elv 3468	. . . . . . . 8 ⊢ [𝑚](𝑅 ⋉ ◡ E ) = ([𝑚]𝑅 × 𝑚)
11	10	uneq2i 4127	. . . . . . 7 ⊢ (𝑚 ∪ [𝑚](𝑅 ⋉ ◡ E )) = (𝑚 ∪ ([𝑚]𝑅 × 𝑚))
12	8, 11	eqtrdi 2820	. . . . . 6 ⊢ (𝑚 ∈ (𝐴 ∖ {∅}) → [𝑚](((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴) = (𝑚 ∪ ([𝑚]𝑅 × 𝑚)))
13	12	eqeq2d 2780	. . . . 5 ⊢ (𝑚 ∈ (𝐴 ∖ {∅}) → (𝑛 = [𝑚](((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴) ↔ 𝑛 = (𝑚 ∪ ([𝑚]𝑅 × 𝑚))))
14	13	pm5.32i 584	. . . 4 ⊢ ((𝑚 ∈ (𝐴 ∖ {∅}) ∧ 𝑛 = [𝑚](((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴)) ↔ (𝑚 ∈ (𝐴 ∖ {∅}) ∧ 𝑛 = (𝑚 ∪ ([𝑚]𝑅 × 𝑚))))
15	5, 14	bitri 278	. . 3 ⊢ ((𝑚 ∈ dom (((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴) ∧ 𝑛 = [𝑚](((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴)) ↔ (𝑚 ∈ (𝐴 ∖ {∅}) ∧ 𝑛 = (𝑚 ∪ ([𝑚]𝑅 × 𝑚))))
16	15	opabbii 5182	. 2 ⊢ {⟨𝑚, 𝑛⟩ ∣ (𝑚 ∈ dom (((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴) ∧ 𝑛 = [𝑚](((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴))} = {⟨𝑚, 𝑛⟩ ∣ (𝑚 ∈ (𝐴 ∖ {∅}) ∧ 𝑛 = (𝑚 ∪ ([𝑚]𝑅 × 𝑚)))}
17	1, 2, 16	3eqtri 2796	1 ⊢ ((𝑅 ⋉ ◡ E ) AdjLiftMap 𝐴) = {⟨𝑚, 𝑛⟩ ∣ (𝑚 ∈ (𝐴 ∖ {∅}) ∧ 𝑛 = (𝑚 ∪ ([𝑚]𝑅 × 𝑚)))}

Syntax hints:   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463   ∖ cdif 3910   ∪ cun 3911  ∅c0 4294  {csn 4594  {copab 5177   ↦ cmpt 5196   E cep 5561   × cxp 5660  ^◡ccnv 5661  dom cdm 5662   ↾ cres 5664  [cec 8691   ⋉ cxrn 38712   AdjLiftMap cadjliftmap 38714

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-eprel 5562  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fo 6543  df-fv 6545  df-oprab 7415  df-1st 7985  df-2nd 7986  df-ec 8695  df-xrn 38918  df-qmap 38984  df-adjliftmap 38993

This theorem is referenced by: (None)