Theorem dfblockliftmap2 39210

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 39209	. 2 ⊢ (𝑅 BlockLiftMap 𝐴) = (𝑚 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ↦ [𝑚](𝑅 ⋉ (◡ E ↾ 𝐴)))
2		elinel1 4147	. . . . 5 ⊢ (𝑚 ∈ (𝐴 ∩ (dom 𝑅 ∖ {∅})) → 𝑚 ∈ 𝐴)
3		dmxrncnvepres2 39182	. . . . 5 ⊢ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) = (𝐴 ∩ (dom 𝑅 ∖ {∅}))
4	2, 3	eleq2s 2878	. . . 4 ⊢ (𝑚 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) → 𝑚 ∈ 𝐴)
5		xrnres2 39175	. . . . . . 7 ⊢ ((𝑅 ⋉ ◡ E ) ↾ 𝐴) = (𝑅 ⋉ (◡ E ↾ 𝐴))
6	5	eceq2i 8740	. . . . . 6 ⊢ [𝑚]((𝑅 ⋉ ◡ E ) ↾ 𝐴) = [𝑚](𝑅 ⋉ (◡ E ↾ 𝐴))
7		elecreseq 8747	. . . . . 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 39159	. . . . 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 5200	. 2 ⊢ (𝑚 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ↦ [𝑚](𝑅 ⋉ (◡ E ↾ 𝐴))) = (𝑚 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ↦ ([𝑚]𝑅 × 𝑚))
13	3	mpteq1i 5196	. 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 1570   ∈ wcel 2145   ∖ cdif 3896   ∩ cin 3898  ∅c0 4279  {csn 4584   ↦ cmpt 5186   E cep 5554   × cxp 5653  ^◡ccnv 5654  dom cdm 5655   ↾ cres 5657  [cec 8695   ⋉ cxrn 38923   BlockLiftMap cblockliftmap 38926

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-eprel 5555  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fo 6539  df-fv 6541  df-oprab 7418  df-1st 7987  df-2nd 7988  df-ec 8699  df-xrn 39129  df-qmap 39195  df-blockliftmap 39208

This theorem is used by: (None)