Theorem dfblockliftfix2 38836

Description: Alternate definition of the equilibrium / fixed-point condition for "block carriers", cf. df-blockliftfix 38594. (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 38835	. . . 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 5163	. 2 ⊢ {⟨𝑟, 𝑎⟩ ∣ (𝑟 ∈ Rels ∧ (𝑟 ⋉ (◡ E ↾ 𝑎)) DomainQs 𝑎)} = {⟨𝑟, 𝑎⟩ ∣ (𝑟 ∈ Rels ∧ (dom (𝑟 ⋉ (◡ E ↾ 𝑎)) / (𝑟 ⋉ (◡ E ↾ 𝑎))) = 𝑎)}
4		resopab 5991	. 2 ⊢ ({⟨𝑟, 𝑎⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑎)) DomainQs 𝑎} ↾ Rels ) = {⟨𝑟, 𝑎⟩ ∣ (𝑟 ∈ Rels ∧ (𝑟 ⋉ (◡ E ↾ 𝑎)) DomainQs 𝑎)}
5		df-blockliftfix 38594	. 2 ⊢ BlockLiftFix = {⟨𝑟, 𝑎⟩ ∣ (𝑟 ∈ Rels ∧ (dom (𝑟 ⋉ (◡ E ↾ 𝑎)) / (𝑟 ⋉ (◡ E ↾ 𝑎))) = 𝑎)}
6	3, 4, 5	3eqtr4ri 2768	1 ⊢ BlockLiftFix = ({⟨𝑟, 𝑎⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑎)) DomainQs 𝑎} ↾ Rels )

Syntax hints:   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {copab 5158   E cep 5521  ^◡ccnv 5621  dom cdm 5622   ↾ cres 5624   / cqs 8632   ⋉ cxrn 38314   BlockLiftFix cblockliftfix 38320   Rels crels 38324   DomainQs wdmqs 38346

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 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

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 2713  df-cleq 2726  df-clel 2809  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-opab 5159  df-xp 5628  df-rel 5629  df-res 5634  df-blockliftfix 38594  df-dmqs 38835

This theorem is referenced by: (None)