Theorem dfadjliftmap2 39389

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 39388	. 2 ⊢ (𝑅 AdjLiftMap 𝐴) = (𝑚 ∈ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) ↦ [𝑚]((𝑅 ∪ ◡ E ) ↾ 𝐴))
2		elinel1 4147	. . . . 5 ⊢ (𝑚 ∈ (𝐴 ∩ (dom 𝑅 ∪ (V ∖ {∅}))) → 𝑚 ∈ 𝐴)
3		dmuncnvepres 39323	. . . . 5 ⊢ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) = (𝐴 ∩ (dom 𝑅 ∪ (V ∖ {∅})))
4	2, 3	eleq2s 2879	. . . 4 ⊢ (𝑚 ∈ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) → 𝑚 ∈ 𝐴)
5		ecuncnvepres 39327	. . . 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 5200	. 2 ⊢ (𝑚 ∈ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) ↦ [𝑚]((𝑅 ∪ ◡ E ) ↾ 𝐴)) = (𝑚 ∈ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) ↦ (𝑚 ∪ [𝑚]𝑅))
8	3	mpteq1i 5196	. 2 ⊢ (𝑚 ∈ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) ↦ (𝑚 ∪ [𝑚]𝑅)) = (𝑚 ∈ (𝐴 ∩ (dom 𝑅 ∪ (V ∖ {∅}))) ↦ (𝑚 ∪ [𝑚]𝑅))
9	1, 7, 8	3eqtri 2788	1 ⊢ (𝑅 AdjLiftMap 𝐴) = (𝑚 ∈ (𝐴 ∩ (dom 𝑅 ∪ (V ∖ {∅}))) ↦ (𝑚 ∪ [𝑚]𝑅))

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145  Vcvv 3451   ∖ cdif 3896   ∪ cun 3897   ∩ cin 3898  ∅c0 4279  {csn 4584   ↦ cmpt 5186   E cep 5550  ^◡ccnv 5650  dom cdm 5651   ↾ cres 5653  [cec 8715   AdjLiftMap cadjliftmap 39108

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-8 2147  ax-9 2155  ax-10 2178  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391

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 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-mpt 5187  df-eprel 5551  df-xp 5657  df-rel 5658  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ec 8719  df-qmap 39378  df-adjliftmap 39387

This theorem is used by: (None)