Theorem dfadjliftmap2 38795

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 38794	. 2 ⊢ (𝑅 AdjLiftMap 𝐴) = (𝑚 ∈ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) ↦ [𝑚]((𝑅 ∪ ◡ E ) ↾ 𝐴))
2		elinel1 4142	. . . . 5 ⊢ (𝑚 ∈ (𝐴 ∩ (dom 𝑅 ∪ (V ∖ {∅}))) → 𝑚 ∈ 𝐴)
3		dmuncnvepres 38729	. . . . 5 ⊢ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) = (𝐴 ∩ (dom 𝑅 ∪ (V ∖ {∅})))
4	2, 3	eleq2s 2855	. . . 4 ⊢ (𝑚 ∈ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) → 𝑚 ∈ 𝐴)
5		ecuncnvepres 38733	. . . 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 5181	. 2 ⊢ (𝑚 ∈ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) ↦ [𝑚]((𝑅 ∪ ◡ E ) ↾ 𝐴)) = (𝑚 ∈ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) ↦ (𝑚 ∪ [𝑚]𝑅))
8	3	mpteq1i 5177	. 2 ⊢ (𝑚 ∈ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) ↦ (𝑚 ∪ [𝑚]𝑅)) = (𝑚 ∈ (𝐴 ∩ (dom 𝑅 ∪ (V ∖ {∅}))) ↦ (𝑚 ∪ [𝑚]𝑅))
9	1, 7, 8	3eqtri 2764	1 ⊢ (𝑅 AdjLiftMap 𝐴) = (𝑚 ∈ (𝐴 ∩ (dom 𝑅 ∪ (V ∖ {∅}))) ↦ (𝑚 ∪ [𝑚]𝑅))

Syntax hints:   = wceq 1542   ∈ wcel 2114  Vcvv 3430   ∖ cdif 3887   ∪ cun 3888   ∩ cin 3889  ∅c0 4274  {csn 4568   ↦ cmpt 5167   E cep 5524  ^◡ccnv 5624  dom cdm 5625   ↾ cres 5627  [cec 8635   AdjLiftMap cadjliftmap 38514

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-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-mpt 5168  df-eprel 5525  df-xp 5631  df-rel 5632  df-cnv 5633  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ec 8639  df-qmap 38784  df-adjliftmap 38793

This theorem is referenced by: (None)