Theorem dfadjliftmap2 39433

Description: Alternate definition of the adjoined lift map. (Contributed by Peter Mazsa, 28-Jan-2026.)

Assertion

Ref	Expression
dfadjliftmap2	⊢ (𝑅 AdjLiftMap 𝐴) = (𝑚 ∈ (𝐴 ∩ (dom 𝑅 ∪ (V ∖ {∅}))) ↦ (𝑚 ∪ [𝑚]𝑅))

Distinct variable groups:   𝐴,𝑚   𝑅,𝑚

Proof of Theorem dfadjliftmap2

Step	Hyp	Ref	Expression
1		dfadjliftmap 39432	. 2 ⊢ (𝑅 AdjLiftMap 𝐴) = (𝑚 ∈ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) ↦ [𝑚]((𝑅 ∪ ◡ E ) ↾ 𝐴))
2		elinel1 4098	. . . . 5 ⊢ (𝑚 ∈ (𝐴 ∩ (dom 𝑅 ∪ (V ∖ {∅}))) → 𝑚 ∈ 𝐴)
3		dmuncnvepres 39367	. . . . 5 ⊢ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) = (𝐴 ∩ (dom 𝑅 ∪ (V ∖ {∅})))
4	2, 3	eleq2s 2878	. . . 4 ⊢ (𝑚 ∈ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) → 𝑚 ∈ 𝐴)
5		ecuncnvepres 39371	. . . 4 ⊢ (𝑚 ∈ 𝐴 → [𝑚]((𝑅 ∪ ◡ E ) ↾ 𝐴) = (𝑚 ∪ [𝑚]𝑅))
6	4, 5	syl 18	. . 3 ⊢ (𝑚 ∈ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) → [𝑚]((𝑅 ∪ ◡ E ) ↾ 𝐴) = (𝑚 ∪ [𝑚]𝑅))
7	6	mpteq2ia 5199	. 2 ⊢ (𝑚 ∈ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) ↦ [𝑚]((𝑅 ∪ ◡ E ) ↾ 𝐴)) = (𝑚 ∈ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) ↦ (𝑚 ∪ [𝑚]𝑅))
8	3	mpteq1i 5195	. 2 ⊢ (𝑚 ∈ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) ↦ (𝑚 ∪ [𝑚]𝑅)) = (𝑚 ∈ (𝐴 ∩ (dom 𝑅 ∪ (V ∖ {∅}))) ↦ (𝑚 ∪ [𝑚]𝑅))
9	1, 7, 8	3eqtri 2787	1 ⊢ (𝑅 AdjLiftMap 𝐴) = (𝑚 ∈ (𝐴 ∩ (dom 𝑅 ∪ (V ∖ {∅}))) ↦ (𝑚 ∪ [𝑚]𝑅))

This proof depends on syntax axioms:   = wceq 1569   ∈ wcel 2144  Vcvv 3450   ∖ cdif 4045   ∩ cin 4089   ∪ cun 4149  ∅c0 4278  {csn 4583   ↦ cmpt 5185   E cep 5549  ^◡ccnv 5649  dom cdm 5650   ↾ cres 5652  [cec 8716   AdjLiftMap cadjliftmap 39152

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-mpt 5186  df-eprel 5550  df-xp 5656  df-rel 5657  df-cnv 5658  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ec 8720  df-qmap 39422  df-adjliftmap 39431

This theorem is used by: (None)