Theorem dfblockliftmap2 38635

Description: Alternate definition of the block lift map. (Contributed by Peter Mazsa, 29-Jan-2026.)

Assertion

Ref	Expression
dfblockliftmap2	⊢ (𝑅 BlockLiftMap 𝐴) = (𝑚 ∈ (𝐴 ∩ (dom 𝑅 ∖ {∅})) ↦ ([𝑚]𝑅 × 𝑚))

Distinct variable groups:   𝐴,𝑚   𝑅,𝑚

Proof of Theorem dfblockliftmap2

Step	Hyp	Ref	Expression
1		df-blockliftmap 38634	. 2 ⊢ (𝑅 BlockLiftMap 𝐴) = (𝑚 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ↦ [𝑚](𝑅 ⋉ (◡ E ↾ 𝐴)))
2		elinel1 4153	. . . . 5 ⊢ (𝑚 ∈ (𝐴 ∩ (dom 𝑅 ∖ {∅})) → 𝑚 ∈ 𝐴)
3		dmxrncnvepres2 38618	. . . . 5 ⊢ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) = (𝐴 ∩ (dom 𝑅 ∖ {∅}))
4	2, 3	eleq2s 2854	. . . 4 ⊢ (𝑚 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) → 𝑚 ∈ 𝐴)
5		xrnres2 38611	. . . . . . 7 ⊢ ((𝑅 ⋉ ◡ E ) ↾ 𝐴) = (𝑅 ⋉ (◡ E ↾ 𝐴))
6	5	eceq2i 8677	. . . . . 6 ⊢ [𝑚]((𝑅 ⋉ ◡ E ) ↾ 𝐴) = [𝑚](𝑅 ⋉ (◡ E ↾ 𝐴))
7		elecreseq 8684	. . . . . 6 ⊢ (𝑚 ∈ 𝐴 → [𝑚]((𝑅 ⋉ ◡ E ) ↾ 𝐴) = [𝑚](𝑅 ⋉ ◡ E ))
8	6, 7	eqtr3id 2785	. . . . 5 ⊢ (𝑚 ∈ 𝐴 → [𝑚](𝑅 ⋉ (◡ E ↾ 𝐴)) = [𝑚](𝑅 ⋉ ◡ E ))
9		ecxrncnvep2 38595	. . . . 5 ⊢ (𝑚 ∈ 𝐴 → [𝑚](𝑅 ⋉ ◡ E ) = ([𝑚]𝑅 × 𝑚))
10	8, 9	eqtrd 2771	. . . 4 ⊢ (𝑚 ∈ 𝐴 → [𝑚](𝑅 ⋉ (◡ E ↾ 𝐴)) = ([𝑚]𝑅 × 𝑚))
11	4, 10	syl 17	. . 3 ⊢ (𝑚 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) → [𝑚](𝑅 ⋉ (◡ E ↾ 𝐴)) = ([𝑚]𝑅 × 𝑚))
12	11	mpteq2ia 5193	. 2 ⊢ (𝑚 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ↦ [𝑚](𝑅 ⋉ (◡ E ↾ 𝐴))) = (𝑚 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ↦ ([𝑚]𝑅 × 𝑚))
13	3	mpteq1i 5189	. 2 ⊢ (𝑚 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ↦ ([𝑚]𝑅 × 𝑚)) = (𝑚 ∈ (𝐴 ∩ (dom 𝑅 ∖ {∅})) ↦ ([𝑚]𝑅 × 𝑚))
14	1, 12, 13	3eqtri 2763	1 ⊢ (𝑅 BlockLiftMap 𝐴) = (𝑚 ∈ (𝐴 ∩ (dom 𝑅 ∖ {∅})) ↦ ([𝑚]𝑅 × 𝑚))

Syntax hints:   = wceq 1541   ∈ wcel 2113   ∖ cdif 3898   ∩ cin 3900  ∅c0 4285  {csn 4580   ↦ cmpt 5179   E cep 5523   × cxp 5622  ^◡ccnv 5623  dom cdm 5624   ↾ cres 5626  [cec 8633   ⋉ cxrn 38375   BlockLiftMap cblockliftmap 38377

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  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 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-eprel 5524  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fo 6498  df-fv 6500  df-oprab 7362  df-1st 7933  df-2nd 7934  df-ec 8637  df-xrn 38565  df-blockliftmap 38634

This theorem is referenced by: (None)