Theorem eceldmqsxrncnvepres2 38933

Description: An (𝑅 ⋉ (^◡ E ↾ 𝐴))-coset in its domain quotient. In the pet 39461 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 38927	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑋) → (𝑅 ⋉ (◡ E ↾ 𝐴)) ∈ V)
2		eceldmqs 8769	. . . 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 17	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑋) → ([𝐵](𝑅 ⋉ (◡ E ↾ 𝐴)) ∈ (dom (𝑅 ⋉ (◡ E ↾ 𝐴)) / (𝑅 ⋉ (◡ E ↾ 𝐴))) ↔ 𝐵 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))))
4	3	3adant2 1144	. 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 38931	. . 3 ⊢ (𝐵 ∈ 𝑊 → (𝐵 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ↔ (𝐵 ∈ 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝐵 ∧ ∃𝑦 𝐵𝑅𝑦)))
6	5	3ad2ant2 1147	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑅 ∈ 𝑋) → (𝐵 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ↔ (𝐵 ∈ 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝐵 ∧ ∃𝑦 𝐵𝑅𝑦)))
7	4, 6	bitrd 281	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)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1098  ∃wex 1799   ∈ wcel 2142  Vcvv 3454   class class class wbr 5100   E cep 5546  ^◡ccnv 5646  dom cdm 5647   ↾ cres 5649  [cec 8676   / cqs 8677   ⋉ cxrn 38670

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-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  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-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  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-qs 8684  df-xrn 38876

This theorem is referenced by: (None)