Theorem dfadjliftmap 39055

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 39054	. 2 ⊢ (𝑅 AdjLiftMap 𝐴) = QMap ((𝑅 ∪ ◡ E ) ↾ 𝐴)
2		df-qmap 39045	. 2 ⊢ QMap ((𝑅 ∪ ◡ E ) ↾ 𝐴) = (𝑚 ∈ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) ↦ [𝑚]((𝑅 ∪ ◡ E ) ↾ 𝐴))
3	1, 2	eqtri 2793	1 ⊢ (𝑅 AdjLiftMap 𝐴) = (𝑚 ∈ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) ↦ [𝑚]((𝑅 ∪ ◡ E ) ↾ 𝐴))

Syntax hints:   = wceq 1568   ∪ cun 3911   ↦ cmpt 5197   E cep 5564  ^◡ccnv 5664  dom cdm 5665   ↾ cres 5667  [cec 8695   QMap cqmap 38774   AdjLiftMap cadjliftmap 38775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-9 2160  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1808  df-cleq 2762  df-qmap 39045  df-adjliftmap 39054

This theorem is referenced by:  dfadjliftmap2  39056  blockadjliftmap  39057  dfsucmap3  39062  dfsucmap2  39063