Theorem dfblockliftmap2 39530

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		dfblockliftmap 39529	. 2 ⊢ (𝑅 BlockLiftMap 𝐴) = (𝑚 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ↦ [𝑚](𝑅 ⋉ (◡ E ↾ 𝐴)))
2		elinel1 4097	. . . . 5 ⊢ (𝑚 ∈ (𝐴 ∩ (dom 𝑅 ∖ {∅})) → 𝑚 ∈ 𝐴)
3		dmxrncnvepres2 39502	. . . . 5 ⊢ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) = (𝐴 ∩ (dom 𝑅 ∖ {∅}))
4	2, 3	eleq2s 2878	. . . 4 ⊢ (𝑚 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) → 𝑚 ∈ 𝐴)
5		xrnres2 39495	. . . . . . 7 ⊢ ((𝑅 ⋉ ◡ E ) ↾ 𝐴) = (𝑅 ⋉ (◡ E ↾ 𝐴))
6	5	eceq2i 8768	. . . . . 6 ⊢ [𝑚]((𝑅 ⋉ ◡ E ) ↾ 𝐴) = [𝑚](𝑅 ⋉ (◡ E ↾ 𝐴))
7		elecreseq 8775	. . . . . 6 ⊢ (𝑚 ∈ 𝐴 → [𝑚]((𝑅 ⋉ ◡ E ) ↾ 𝐴) = [𝑚](𝑅 ⋉ ◡ E ))
8	6, 7	eqtr3id 2809	. . . . 5 ⊢ (𝑚 ∈ 𝐴 → [𝑚](𝑅 ⋉ (◡ E ↾ 𝐴)) = [𝑚](𝑅 ⋉ ◡ E ))
9		ecxrncnvep2 39479	. . . . 5 ⊢ (𝑚 ∈ 𝐴 → [𝑚](𝑅 ⋉ ◡ E ) = ([𝑚]𝑅 × 𝑚))
10	8, 9	eqtrd 2795	. . . 4 ⊢ (𝑚 ∈ 𝐴 → [𝑚](𝑅 ⋉ (◡ E ↾ 𝐴)) = ([𝑚]𝑅 × 𝑚))
11	4, 10	syl 18	. . 3 ⊢ (𝑚 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) → [𝑚](𝑅 ⋉ (◡ E ↾ 𝐴)) = ([𝑚]𝑅 × 𝑚))
12	11	mpteq2ia 5199	. 2 ⊢ (𝑚 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ↦ [𝑚](𝑅 ⋉ (◡ E ↾ 𝐴))) = (𝑚 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ↦ ([𝑚]𝑅 × 𝑚))
13	3	mpteq1i 5195	. 2 ⊢ (𝑚 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ↦ ([𝑚]𝑅 × 𝑚)) = (𝑚 ∈ (𝐴 ∩ (dom 𝑅 ∖ {∅})) ↦ ([𝑚]𝑅 × 𝑚))
14	1, 12, 13	3eqtri 2787	1 ⊢ (𝑅 BlockLiftMap 𝐴) = (𝑚 ∈ (𝐴 ∩ (dom 𝑅 ∖ {∅})) ↦ ([𝑚]𝑅 × 𝑚))

This proof depends on syntax axioms:   = wceq 1569   ∈ wcel 2144   ∖ cdif 4044   ∩ cin 4088  ∅c0 4277  {csn 4584   ↦ cmpt 5185   E cep 5550   × cxp 5649  ^◡ccnv 5650  dom cdm 5651   ↾ cres 5653  [cec 8723   ⋉ cxrn 39243   BlockLiftMap cblockliftmap 39246

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-eprel 5551  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-oprab 7428  df-1st 8005  df-2nd 8006  df-ec 8727  df-xrn 39449  df-qmap 39515  df-blockliftmap 39528

This theorem is used by: (None)