Theorem dfadjliftmap2 38956

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 38955	. 2 ⊢ (𝑅 AdjLiftMap 𝐴) = (𝑚 ∈ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) ↦ [𝑚]((𝑅 ∪ ◡ E ) ↾ 𝐴))
2		elinel1 4153	. . . . 5 ⊢ (𝑚 ∈ (𝐴 ∩ (dom 𝑅 ∪ (V ∖ {∅}))) → 𝑚 ∈ 𝐴)
3		dmuncnvepres 38890	. . . . 5 ⊢ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) = (𝐴 ∩ (dom 𝑅 ∪ (V ∖ {∅})))
4	2, 3	eleq2s 2880	. . . 4 ⊢ (𝑚 ∈ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) → 𝑚 ∈ 𝐴)
5		ecuncnvepres 38894	. . . 4 ⊢ (𝑚 ∈ 𝐴 → [𝑚]((𝑅 ∪ ◡ E ) ↾ 𝐴) = (𝑚 ∪ [𝑚]𝑅))
6	4, 5	syl 17	. . 3 ⊢ (𝑚 ∈ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) → [𝑚]((𝑅 ∪ ◡ E ) ↾ 𝐴) = (𝑚 ∪ [𝑚]𝑅))
7	6	mpteq2ia 5195	. 2 ⊢ (𝑚 ∈ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) ↦ [𝑚]((𝑅 ∪ ◡ E ) ↾ 𝐴)) = (𝑚 ∈ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) ↦ (𝑚 ∪ [𝑚]𝑅))
8	3	mpteq1i 5191	. 2 ⊢ (𝑚 ∈ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) ↦ (𝑚 ∪ [𝑚]𝑅)) = (𝑚 ∈ (𝐴 ∩ (dom 𝑅 ∪ (V ∖ {∅}))) ↦ (𝑚 ∪ [𝑚]𝑅))
9	1, 7, 8	3eqtri 2789	1 ⊢ (𝑅 AdjLiftMap 𝐴) = (𝑚 ∈ (𝐴 ∩ (dom 𝑅 ∪ (V ∖ {∅}))) ↦ (𝑚 ∪ [𝑚]𝑅))

Syntax hints:   = wceq 1560   ∈ wcel 2142  Vcvv 3454   ∖ cdif 3901   ∪ cun 3902   ∩ cin 3903  ∅c0 4285  {csn 4582   ↦ cmpt 5181   E cep 5546  ^◡ccnv 5646  dom cdm 5647   ↾ cres 5649  [cec 8676   AdjLiftMap cadjliftmap 38675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-mpt 5182  df-eprel 5547  df-xp 5653  df-rel 5654  df-cnv 5655  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-ec 8680  df-qmap 38945  df-adjliftmap 38954

This theorem is referenced by: (None)