Theorem dfblockliftfix2 38897

Description: Alternate definition of the equilibrium / fixed-point condition for "block carriers", cf. df-blockliftfix 38655. (Contributed by Peter Mazsa, 29-Jan-2026.)

Assertion

Ref	Expression
dfblockliftfix2	⊢ BlockLiftFix = ({⟨𝑟, 𝑎⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑎)) DomainQs 𝑎} ↾ Rels )

Distinct variable group:   𝑟,𝑎

Proof of Theorem dfblockliftfix2

Step	Hyp	Ref	Expression
1		df-dmqs 38896	. . . 4 ⊢ ((𝑟 ⋉ (◡ E ↾ 𝑎)) DomainQs 𝑎 ↔ (dom (𝑟 ⋉ (◡ E ↾ 𝑎)) / (𝑟 ⋉ (◡ E ↾ 𝑎))) = 𝑎)
2	1	anbi2i 623	. . 3 ⊢ ((𝑟 ∈ Rels ∧ (𝑟 ⋉ (◡ E ↾ 𝑎)) DomainQs 𝑎) ↔ (𝑟 ∈ Rels ∧ (dom (𝑟 ⋉ (◡ E ↾ 𝑎)) / (𝑟 ⋉ (◡ E ↾ 𝑎))) = 𝑎))
3	2	opabbii 5165	. 2 ⊢ {⟨𝑟, 𝑎⟩ ∣ (𝑟 ∈ Rels ∧ (𝑟 ⋉ (◡ E ↾ 𝑎)) DomainQs 𝑎)} = {⟨𝑟, 𝑎⟩ ∣ (𝑟 ∈ Rels ∧ (dom (𝑟 ⋉ (◡ E ↾ 𝑎)) / (𝑟 ⋉ (◡ E ↾ 𝑎))) = 𝑎)}
4		resopab 5993	. 2 ⊢ ({⟨𝑟, 𝑎⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑎)) DomainQs 𝑎} ↾ Rels ) = {⟨𝑟, 𝑎⟩ ∣ (𝑟 ∈ Rels ∧ (𝑟 ⋉ (◡ E ↾ 𝑎)) DomainQs 𝑎)}
5		df-blockliftfix 38655	. 2 ⊢ BlockLiftFix = {⟨𝑟, 𝑎⟩ ∣ (𝑟 ∈ Rels ∧ (dom (𝑟 ⋉ (◡ E ↾ 𝑎)) / (𝑟 ⋉ (◡ E ↾ 𝑎))) = 𝑎)}
6	3, 4, 5	3eqtr4ri 2770	1 ⊢ BlockLiftFix = ({⟨𝑟, 𝑎⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑎)) DomainQs 𝑎} ↾ Rels )

Syntax hints:   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {copab 5160   E cep 5523  ^◡ccnv 5623  dom cdm 5624   ↾ cres 5626   / cqs 8634   ⋉ cxrn 38375   BlockLiftFix cblockliftfix 38381   Rels crels 38385   DomainQs wdmqs 38407

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-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

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-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  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-opab 5161  df-xp 5630  df-rel 5631  df-res 5636  df-blockliftfix 38655  df-dmqs 38896

This theorem is referenced by: (None)