Theorem eceldmqsxrncnvepres2 39144

Description: An (𝑅 ⋉ (^◡ E ↾ 𝐴))-coset in its domain quotient. In the pet 39672 span (𝑅 ⋉ (^◡ E ↾ 𝐴)), a block [ B ] lies in the domain quotient exactly when its representative 𝐵 belongs to 𝐴 and actually fires at least one arrow (has some 𝑥 ∈ 𝐵 and some 𝑦 with 𝐵𝑅𝑦). (Contributed by Peter Mazsa, 23-Nov-2025.)

Assertion

Ref	Expression
eceldmqsxrncnvepres2	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑅 ∈ 𝑋) → ([𝐵](𝑅 ⋉ (◡ E ↾ 𝐴)) ∈ (dom (𝑅 ⋉ (◡ E ↾ 𝐴)) / (𝑅 ⋉ (◡ E ↾ 𝐴))) ↔ (𝐵 ∈ 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝐵 ∧ ∃𝑦 𝐵𝑅𝑦)))

Distinct variable groups:   𝑦,𝐴   𝑥,𝐵   𝑦,𝐵   𝑦,𝑅

Allowed substitution hints:   𝐴(𝑥)   𝑅(𝑥)   𝑉(𝑥, 𝑦)   𝑊(𝑥, 𝑦)   𝑋(𝑥, 𝑦)

Proof of Theorem eceldmqsxrncnvepres2

Step	Hyp	Ref	Expression
1		xrncnvepresex 39138	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑋) → (𝑅 ⋉ (◡ E ↾ 𝐴)) ∈ V)
2		eceldmqs 8791	. . . 4 ⊢ ((𝑅 ⋉ (◡ E ↾ 𝐴)) ∈ V → ([𝐵](𝑅 ⋉ (◡ E ↾ 𝐴)) ∈ (dom (𝑅 ⋉ (◡ E ↾ 𝐴)) / (𝑅 ⋉ (◡ E ↾ 𝐴))) ↔ 𝐵 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))))
3	1, 2	syl 18	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑋) → ([𝐵](𝑅 ⋉ (◡ E ↾ 𝐴)) ∈ (dom (𝑅 ⋉ (◡ E ↾ 𝐴)) / (𝑅 ⋉ (◡ E ↾ 𝐴))) ↔ 𝐵 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))))
4	3	3adant2 1149	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑅 ∈ 𝑋) → ([𝐵](𝑅 ⋉ (◡ E ↾ 𝐴)) ∈ (dom (𝑅 ⋉ (◡ E ↾ 𝐴)) / (𝑅 ⋉ (◡ E ↾ 𝐴))) ↔ 𝐵 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))))
5		eldmxrncnvepres2 39142	. . 3 ⊢ (𝐵 ∈ 𝑊 → (𝐵 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ↔ (𝐵 ∈ 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝐵 ∧ ∃𝑦 𝐵𝑅𝑦)))
6	5	3ad2ant2 1152	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑅 ∈ 𝑋) → (𝐵 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ↔ (𝐵 ∈ 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝐵 ∧ ∃𝑦 𝐵𝑅𝑦)))
7	4, 6	bitrd 282	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑅 ∈ 𝑋) → ([𝐵](𝑅 ⋉ (◡ E ↾ 𝐴)) ∈ (dom (𝑅 ⋉ (◡ E ↾ 𝐴)) / (𝑅 ⋉ (◡ E ↾ 𝐴))) ↔ (𝐵 ∈ 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝐵 ∧ ∃𝑦 𝐵𝑅𝑦)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∃wex 1812   ∈ wcel 2146  Vcvv 3457   class class class wbr 5111   E cep 5562  ^◡ccnv 5662  dom cdm 5663   ↾ cres 5665  [cec 8698   / cqs 8699   ⋉ cxrn 38881

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-eprel 5563  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fo 6546  df-fv 6548  df-oprab 7423  df-1st 7992  df-2nd 7993  df-ec 8702  df-qs 8706  df-xrn 39087

This theorem is used by: (None)