Theorem dfblockliftfix2 38746

Description: Alternate definition of the equilibrium / fixed-point condition for "block carriers", cf. df-blockliftfix 38504. (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 38745	. . . 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 5156	. 2 ⊢ {⟨𝑟, 𝑎⟩ ∣ (𝑟 ∈ Rels ∧ (𝑟 ⋉ (◡ E ↾ 𝑎)) DomainQs 𝑎)} = {⟨𝑟, 𝑎⟩ ∣ (𝑟 ∈ Rels ∧ (dom (𝑟 ⋉ (◡ E ↾ 𝑎)) / (𝑟 ⋉ (◡ E ↾ 𝑎))) = 𝑎)}
4		resopab 5982	. 2 ⊢ ({⟨𝑟, 𝑎⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑎)) DomainQs 𝑎} ↾ Rels ) = {⟨𝑟, 𝑎⟩ ∣ (𝑟 ∈ Rels ∧ (𝑟 ⋉ (◡ E ↾ 𝑎)) DomainQs 𝑎)}
5		df-blockliftfix 38504	. 2 ⊢ BlockLiftFix = {⟨𝑟, 𝑎⟩ ∣ (𝑟 ∈ Rels ∧ (dom (𝑟 ⋉ (◡ E ↾ 𝑎)) / (𝑟 ⋉ (◡ E ↾ 𝑎))) = 𝑎)}
6	3, 4, 5	3eqtr4ri 2765	1 ⊢ BlockLiftFix = ({⟨𝑟, 𝑎⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑎)) DomainQs 𝑎} ↾ Rels )

Syntax hints:   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {copab 5151   E cep 5513  ^◡ccnv 5613  dom cdm 5614   ↾ cres 5616   / cqs 8621   ⋉ cxrn 38224   BlockLiftFix cblockliftfix 38230   Rels crels 38234   DomainQs wdmqs 38256

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 2113  ax-9 2121  ax-10 2144  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

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 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-opab 5152  df-xp 5620  df-rel 5621  df-res 5626  df-blockliftfix 38504  df-dmqs 38745

This theorem is referenced by: (None)