Theorem blockadjliftmap 39107

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 39105	. 2 ⊢ ((𝑅 ⋉ ◡ E ) AdjLiftMap 𝐴) = (𝑚 ∈ dom (((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴) ↦ [𝑚](((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴))
2		df-mpt 5193	. 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 39041	. . . . . 6 ⊢ dom (((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴) = (𝐴 ∖ {∅})
4	3	eleq2i 2855	. . . . 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 4085	. . . . . . . 8 ⊢ (𝑚 ∈ (𝐴 ∖ {∅}) → 𝑚 ∈ 𝐴)
7		ecuncnvepres 39044	. . . . . . . 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 39059	. . . . . . . . 9 ⊢ (𝑚 ∈ V → [𝑚](𝑅 ⋉ ◡ E ) = ([𝑚]𝑅 × 𝑚))
10	9	elv 3460	. . . . . . . 8 ⊢ [𝑚](𝑅 ⋉ ◡ E ) = ([𝑚]𝑅 × 𝑚)
11	10	uneq2i 4119	. . . . . . 7 ⊢ (𝑚 ∪ [𝑚](𝑅 ⋉ ◡ E )) = (𝑚 ∪ ([𝑚]𝑅 × 𝑚))
12	8, 11	eqtrdi 2814	. . . . . 6 ⊢ (𝑚 ∈ (𝐴 ∖ {∅}) → [𝑚](((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴) = (𝑚 ∪ ([𝑚]𝑅 × 𝑚)))
13	12	eqeq2d 2774	. . . . 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 5178	. 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 2790	1 ⊢ ((𝑅 ⋉ ◡ E ) AdjLiftMap 𝐴) = {⟨𝑚, 𝑛⟩ ∣ (𝑚 ∈ (𝐴 ∖ {∅}) ∧ 𝑛 = (𝑚 ∪ ([𝑚]𝑅 × 𝑚)))}

Syntax hints:   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ∖ cdif 3902   ∪ cun 3903  ∅c0 4286  {csn 4589  {copab 5173   ↦ cmpt 5192   E cep 5560   × cxp 5659  ^◡ccnv 5660  dom cdm 5661   ↾ cres 5663  [cec 8688   ⋉ cxrn 38823   AdjLiftMap cadjliftmap 38825

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-eprel 5561  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544  df-oprab 7414  df-1st 7982  df-2nd 7983  df-ec 8692  df-xrn 39029  df-qmap 39095  df-adjliftmap 39104

This theorem is referenced by: (None)