Theorem dfadjliftmap 38960

Description: Alternate (expanded) definition of the adjoined lift map. (Contributed by Peter Mazsa, 28-Jan-2026.) (Revised by Peter Mazsa, 22-Feb-2026.)

Assertion

Ref	Expression
dfadjliftmap	⊢ (𝑅 AdjLiftMap 𝐴) = (𝑚 ∈ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) ↦ [𝑚]((𝑅 ∪ ◡ E ) ↾ 𝐴))

Distinct variable groups:   𝐴,𝑚   𝑅,𝑚

Proof of Theorem dfadjliftmap

Step	Hyp	Ref	Expression
1		df-adjliftmap 38959	. 2 ⊢ (𝑅 AdjLiftMap 𝐴) = QMap ((𝑅 ∪ ◡ E ) ↾ 𝐴)
2		df-qmap 38950	. 2 ⊢ QMap ((𝑅 ∪ ◡ E ) ↾ 𝐴) = (𝑚 ∈ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) ↦ [𝑚]((𝑅 ∪ ◡ E ) ↾ 𝐴))
3	1, 2	eqtri 2787	1 ⊢ (𝑅 AdjLiftMap 𝐴) = (𝑚 ∈ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) ↦ [𝑚]((𝑅 ∪ ◡ E ) ↾ 𝐴))

Syntax hints:   = wceq 1562   ∪ cun 3904   ↦ cmpt 5183   E cep 5548  ^◡ccnv 5648  dom cdm 5649   ↾ cres 5651  [cec 8678   QMap cqmap 38679   AdjLiftMap cadjliftmap 38680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1802  df-cleq 2756  df-qmap 38950  df-adjliftmap 38959

This theorem is referenced by:  dfadjliftmap2  38961  blockadjliftmap  38962  dfsucmap3  38967  dfsucmap2  38968