Theorem dfadjliftmap 38736

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 38735	. 2 ⊢ (𝑅 AdjLiftMap 𝐴) = QMap ((𝑅 ∪ ◡ E ) ↾ 𝐴)
2		df-qmap 38726	. 2 ⊢ QMap ((𝑅 ∪ ◡ E ) ↾ 𝐴) = (𝑚 ∈ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) ↦ [𝑚]((𝑅 ∪ ◡ E ) ↾ 𝐴))
3	1, 2	eqtri 2760	1 ⊢ (𝑅 AdjLiftMap 𝐴) = (𝑚 ∈ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) ↦ [𝑚]((𝑅 ∪ ◡ E ) ↾ 𝐴))

Syntax hints:   = wceq 1542   ∪ cun 3901   ↦ cmpt 5181   E cep 5533  ^◡ccnv 5633  dom cdm 5634   ↾ cres 5636  [cec 8645   QMap cqmap 38455   AdjLiftMap cadjliftmap 38456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-cleq 2729  df-qmap 38726  df-adjliftmap 38735

This theorem is referenced by:  dfadjliftmap2  38737  blockadjliftmap  38738  dfsucmap3  38743  dfsucmap2  38744