Theorem dfadjliftmap 39191

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 39190	. 2 ⊢ (𝑅 AdjLiftMap 𝐴) = QMap ((𝑅 ∪ ◡ E ) ↾ 𝐴)
2		df-qmap 39181	. 2 ⊢ QMap ((𝑅 ∪ ◡ E ) ↾ 𝐴) = (𝑚 ∈ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) ↦ [𝑚]((𝑅 ∪ ◡ E ) ↾ 𝐴))
3	1, 2	eqtri 2785	1 ⊢ (𝑅 AdjLiftMap 𝐴) = (𝑚 ∈ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) ↦ [𝑚]((𝑅 ∪ ◡ E ) ↾ 𝐴))

This proof depends on syntax axioms:   = wceq 1570   ∪ cun 3900   ↦ cmpt 5190   E cep 5558  ^◡ccnv 5658  dom cdm 5659   ↾ cres 5661  [cec 8697   QMap cqmap 38910   AdjLiftMap cadjliftmap 38911

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 1943  ax-6 2000  ax-7 2041  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2754  df-qmap 39181  df-adjliftmap 39190

This theorem is used by:  dfadjliftmap2  39192  blockadjliftmap  39193  dfsucmap3  39198  dfsucmap2  39199