Theorem dfblockliftmap2 38960

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 38959	. 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 38932	. . . . 5 ⊢ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) = (𝐴 ∩ (dom 𝑅 ∖ {∅}))
4	2, 3	eleq2s 2880	. . . 4 ⊢ (𝑚 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) → 𝑚 ∈ 𝐴)
5		xrnres2 38925	. . . . . . 7 ⊢ ((𝑅 ⋉ ◡ E ) ↾ 𝐴) = (𝑅 ⋉ (◡ E ↾ 𝐴))
6	5	eceq2i 8721	. . . . . 6 ⊢ [𝑚]((𝑅 ⋉ ◡ E ) ↾ 𝐴) = [𝑚](𝑅 ⋉ (◡ E ↾ 𝐴))
7		elecreseq 8728	. . . . . 6 ⊢ (𝑚 ∈ 𝐴 → [𝑚]((𝑅 ⋉ ◡ E ) ↾ 𝐴) = [𝑚](𝑅 ⋉ ◡ E ))
8	6, 7	eqtr3id 2811	. . . . 5 ⊢ (𝑚 ∈ 𝐴 → [𝑚](𝑅 ⋉ (◡ E ↾ 𝐴)) = [𝑚](𝑅 ⋉ ◡ E ))
9		ecxrncnvep2 38909	. . . . 5 ⊢ (𝑚 ∈ 𝐴 → [𝑚](𝑅 ⋉ ◡ E ) = ([𝑚]𝑅 × 𝑚))
10	8, 9	eqtrd 2797	. . . 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 5195	. 2 ⊢ (𝑚 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ↦ [𝑚](𝑅 ⋉ (◡ E ↾ 𝐴))) = (𝑚 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ↦ ([𝑚]𝑅 × 𝑚))
13	3	mpteq1i 5191	. 2 ⊢ (𝑚 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ↦ ([𝑚]𝑅 × 𝑚)) = (𝑚 ∈ (𝐴 ∩ (dom 𝑅 ∖ {∅})) ↦ ([𝑚]𝑅 × 𝑚))
14	1, 12, 13	3eqtri 2789	1 ⊢ (𝑅 BlockLiftMap 𝐴) = (𝑚 ∈ (𝐴 ∩ (dom 𝑅 ∖ {∅})) ↦ ([𝑚]𝑅 × 𝑚))

Syntax hints:   = wceq 1560   ∈ wcel 2142   ∖ cdif 3901   ∩ cin 3903  ∅c0 4285  {csn 4582   ↦ cmpt 5181   E cep 5546   × cxp 5645  ^◡ccnv 5646  dom cdm 5647   ↾ cres 5649  [cec 8676   ⋉ cxrn 38673   BlockLiftMap cblockliftmap 38676

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-blockliftmap 38958

This theorem is referenced by: (None)