Theorem blockadjliftmap 38779

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 38777	. 2 ⊢ ((𝑅 ⋉ ◡ E ) AdjLiftMap 𝐴) = (𝑚 ∈ dom (((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴) ↦ [𝑚](((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴))
2		df-mpt 5167	. 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 38713	. . . . . 6 ⊢ dom (((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴) = (𝐴 ∖ {∅})
4	3	eleq2i 2828	. . . . 5 ⊢ (𝑚 ∈ dom (((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴) ↔ 𝑚 ∈ (𝐴 ∖ {∅}))
5	4	anbi1i 625	. . . 4 ⊢ ((𝑚 ∈ dom (((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴) ∧ 𝑛 = [𝑚](((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴)) ↔ (𝑚 ∈ (𝐴 ∖ {∅}) ∧ 𝑛 = [𝑚](((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴)))
6		eldifi 4071	. . . . . . . 8 ⊢ (𝑚 ∈ (𝐴 ∖ {∅}) → 𝑚 ∈ 𝐴)
7		ecuncnvepres 38716	. . . . . . . 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 38731	. . . . . . . . 9 ⊢ (𝑚 ∈ V → [𝑚](𝑅 ⋉ ◡ E ) = ([𝑚]𝑅 × 𝑚))
10	9	elv 3434	. . . . . . . 8 ⊢ [𝑚](𝑅 ⋉ ◡ E ) = ([𝑚]𝑅 × 𝑚)
11	10	uneq2i 4105	. . . . . . 7 ⊢ (𝑚 ∪ [𝑚](𝑅 ⋉ ◡ E )) = (𝑚 ∪ ([𝑚]𝑅 × 𝑚))
12	8, 11	eqtrdi 2787	. . . . . 6 ⊢ (𝑚 ∈ (𝐴 ∖ {∅}) → [𝑚](((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴) = (𝑚 ∪ ([𝑚]𝑅 × 𝑚)))
13	12	eqeq2d 2747	. . . . 5 ⊢ (𝑚 ∈ (𝐴 ∖ {∅}) → (𝑛 = [𝑚](((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴) ↔ 𝑛 = (𝑚 ∪ ([𝑚]𝑅 × 𝑚))))
14	13	pm5.32i 574	. . . 4 ⊢ ((𝑚 ∈ (𝐴 ∖ {∅}) ∧ 𝑛 = [𝑚](((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴)) ↔ (𝑚 ∈ (𝐴 ∖ {∅}) ∧ 𝑛 = (𝑚 ∪ ([𝑚]𝑅 × 𝑚))))
15	5, 14	bitri 275	. . 3 ⊢ ((𝑚 ∈ dom (((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴) ∧ 𝑛 = [𝑚](((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴)) ↔ (𝑚 ∈ (𝐴 ∖ {∅}) ∧ 𝑛 = (𝑚 ∪ ([𝑚]𝑅 × 𝑚))))
16	15	opabbii 5152	. 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 2763	1 ⊢ ((𝑅 ⋉ ◡ E ) AdjLiftMap 𝐴) = {⟨𝑚, 𝑛⟩ ∣ (𝑚 ∈ (𝐴 ∖ {∅}) ∧ 𝑛 = (𝑚 ∪ ([𝑚]𝑅 × 𝑚)))}

Syntax hints:   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3429   ∖ cdif 3886   ∪ cun 3887  ∅c0 4273  {csn 4567  {copab 5147   ↦ cmpt 5166   E cep 5530   × cxp 5629  ^◡ccnv 5630  dom cdm 5631   ↾ cres 5633  [cec 8641   ⋉ cxrn 38495   AdjLiftMap cadjliftmap 38497

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-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

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-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-eprel 5531  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fo 6504  df-fv 6506  df-oprab 7371  df-1st 7942  df-2nd 7943  df-ec 8645  df-xrn 38701  df-qmap 38767  df-adjliftmap 38776

This theorem is referenced by: (None)