Theorem blockadjliftmap 38957

Description: A "two-stage" construction is obtained by first forming the block relation (𝑅 ⋉ ^◡ E ) and then adjoining elements as "BlockAdj". Combined, it uses the relation ((𝑅 ⋉ ^◡ E ) ∪ ^◡ E ). (Contributed by Peter Mazsa, 28-Jan-2026.)

Assertion

Ref	Expression
blockadjliftmap	⊢ ((𝑅 ⋉ ◡ E ) AdjLiftMap 𝐴) = {⟨𝑚, 𝑛⟩ ∣ (𝑚 ∈ (𝐴 ∖ {∅}) ∧ 𝑛 = (𝑚 ∪ ([𝑚]𝑅 × 𝑚)))}

Distinct variable groups:   𝐴,𝑚,𝑛   𝑅,𝑚,𝑛

Proof of Theorem blockadjliftmap

Step	Hyp	Ref	Expression
1		dfadjliftmap 38955	. 2 ⊢ ((𝑅 ⋉ ◡ E ) AdjLiftMap 𝐴) = (𝑚 ∈ dom (((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴) ↦ [𝑚](((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴))
2		df-mpt 5182	. 2 ⊢ (𝑚 ∈ dom (((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴) ↦ [𝑚](((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴)) = {⟨𝑚, 𝑛⟩ ∣ (𝑚 ∈ dom (((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴) ∧ 𝑛 = [𝑚](((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴))}
3		dmxrnuncnvepres 38891	. . . . . 6 ⊢ dom (((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴) = (𝐴 ∖ {∅})
4	3	eleq2i 2854	. . . . 5 ⊢ (𝑚 ∈ dom (((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴) ↔ 𝑚 ∈ (𝐴 ∖ {∅}))
5	4	anbi1i 633	. . . 4 ⊢ ((𝑚 ∈ dom (((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴) ∧ 𝑛 = [𝑚](((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴)) ↔ (𝑚 ∈ (𝐴 ∖ {∅}) ∧ 𝑛 = [𝑚](((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴)))
6		eldifi 4084	. . . . . . . 8 ⊢ (𝑚 ∈ (𝐴 ∖ {∅}) → 𝑚 ∈ 𝐴)
7		ecuncnvepres 38894	. . . . . . . 8 ⊢ (𝑚 ∈ 𝐴 → [𝑚](((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴) = (𝑚 ∪ [𝑚](𝑅 ⋉ ◡ E )))
8	6, 7	syl 17	. . . . . . 7 ⊢ (𝑚 ∈ (𝐴 ∖ {∅}) → [𝑚](((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴) = (𝑚 ∪ [𝑚](𝑅 ⋉ ◡ E )))
9		ecxrncnvep2 38909	. . . . . . . . 9 ⊢ (𝑚 ∈ V → [𝑚](𝑅 ⋉ ◡ E ) = ([𝑚]𝑅 × 𝑚))
10	9	elv 3459	. . . . . . . 8 ⊢ [𝑚](𝑅 ⋉ ◡ E ) = ([𝑚]𝑅 × 𝑚)
11	10	uneq2i 4118	. . . . . . 7 ⊢ (𝑚 ∪ [𝑚](𝑅 ⋉ ◡ E )) = (𝑚 ∪ ([𝑚]𝑅 × 𝑚))
12	8, 11	eqtrdi 2813	. . . . . 6 ⊢ (𝑚 ∈ (𝐴 ∖ {∅}) → [𝑚](((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴) = (𝑚 ∪ ([𝑚]𝑅 × 𝑚)))
13	12	eqeq2d 2773	. . . . 5 ⊢ (𝑚 ∈ (𝐴 ∖ {∅}) → (𝑛 = [𝑚](((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴) ↔ 𝑛 = (𝑚 ∪ ([𝑚]𝑅 × 𝑚))))
14	13	pm5.32i 582	. . . 4 ⊢ ((𝑚 ∈ (𝐴 ∖ {∅}) ∧ 𝑛 = [𝑚](((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴)) ↔ (𝑚 ∈ (𝐴 ∖ {∅}) ∧ 𝑛 = (𝑚 ∪ ([𝑚]𝑅 × 𝑚))))
15	5, 14	bitri 277	. . 3 ⊢ ((𝑚 ∈ dom (((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴) ∧ 𝑛 = [𝑚](((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴)) ↔ (𝑚 ∈ (𝐴 ∖ {∅}) ∧ 𝑛 = (𝑚 ∪ ([𝑚]𝑅 × 𝑚))))
16	15	opabbii 5167	. 2 ⊢ {⟨𝑚, 𝑛⟩ ∣ (𝑚 ∈ dom (((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴) ∧ 𝑛 = [𝑚](((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴))} = {⟨𝑚, 𝑛⟩ ∣ (𝑚 ∈ (𝐴 ∖ {∅}) ∧ 𝑛 = (𝑚 ∪ ([𝑚]𝑅 × 𝑚)))}
17	1, 2, 16	3eqtri 2789	1 ⊢ ((𝑅 ⋉ ◡ E ) AdjLiftMap 𝐴) = {⟨𝑚, 𝑛⟩ ∣ (𝑚 ∈ (𝐴 ∖ {∅}) ∧ 𝑛 = (𝑚 ∪ ([𝑚]𝑅 × 𝑚)))}

Syntax hints:   ∧ wa 399   = wceq 1560   ∈ wcel 2142  Vcvv 3454   ∖ cdif 3901   ∪ cun 3902  ∅c0 4285  {csn 4582  {copab 5162   ↦ cmpt 5181   E cep 5546   × cxp 5645  ^◡ccnv 5646  dom cdm 5647   ↾ cres 5649  [cec 8676   ⋉ cxrn 38673   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-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

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-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-sbc 3745  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-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-eprel 5547  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-fo 6527  df-fv 6529  df-oprab 7400  df-1st 7970  df-2nd 7971  df-ec 8680  df-xrn 38879  df-qmap 38945  df-adjliftmap 38954

This theorem is referenced by: (None)