Theorem dfadjliftmap2 39655

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 39654	. 2 ⊢ (𝑅 AdjLiftMap 𝐴) = (𝑚 ∈ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) ↦ [𝑚]((𝑅 ∪ ◡ E ) ↾ 𝐴))
2		elinel1 4101	. . . . 5 ⊢ (𝑚 ∈ (𝐴 ∩ (dom 𝑅 ∪ (V ∖ {∅}))) → 𝑚 ∈ 𝐴)
3		dmuncnvepres 39589	. . . . 5 ⊢ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) = (𝐴 ∩ (dom 𝑅 ∪ (V ∖ {∅})))
4	2, 3	eleq2s 2882	. . . 4 ⊢ (𝑚 ∈ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) → 𝑚 ∈ 𝐴)
5		ecuncnvepres 39593	. . . 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 5206	. 2 ⊢ (𝑚 ∈ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) ↦ [𝑚]((𝑅 ∪ ◡ E ) ↾ 𝐴)) = (𝑚 ∈ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) ↦ (𝑚 ∪ [𝑚]𝑅))
8	3	mpteq1i 5202	. 2 ⊢ (𝑚 ∈ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) ↦ (𝑚 ∪ [𝑚]𝑅)) = (𝑚 ∈ (𝐴 ∩ (dom 𝑅 ∪ (V ∖ {∅}))) ↦ (𝑚 ∪ [𝑚]𝑅))
9	1, 7, 8	3eqtri 2790	1 ⊢ (𝑅 AdjLiftMap 𝐴) = (𝑚 ∈ (𝐴 ∩ (dom 𝑅 ∪ (V ∖ {∅}))) ↦ (𝑚 ∪ [𝑚]𝑅))

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ∖ cdif 4048   ∩ cin 4092   ∪ cun 4152  ∅c0 4281  {csn 4588   ↦ cmpt 5192   E cep 5557  ^◡ccnv 5657  dom cdm 5658   ↾ cres 5660  [cec 8753   AdjLiftMap cadjliftmap 39374

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-mpt 5193  df-eprel 5558  df-xp 5664  df-rel 5665  df-cnv 5666  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-ec 8757  df-qmap 39644  df-adjliftmap 39653

This theorem is used by: (None)