Theorem disjlem18

Description: Lemma for disjdmqseq , partim2 and petlem via disjlem19 , (general version of the former prtlem18 ). (Contributed by Peter Mazsa, 16-Sep-2021)

	Ref	Expression
Assertion	disjlem18	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( Disj 𝑅 → ( ( 𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [ 𝑥 ] 𝑅 ) → ( 𝐵 ∈ [ 𝑥 ] 𝑅 ↔ 𝐴 ≀ 𝑅 𝐵 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		rspe	⊢ ( ( 𝑥 ∈ dom 𝑅 ∧ ( 𝐴 ∈ [ 𝑥 ] 𝑅 ∧ 𝐵 ∈ [ 𝑥 ] 𝑅 ) ) → ∃ 𝑥 ∈ dom 𝑅 ( 𝐴 ∈ [ 𝑥 ] 𝑅 ∧ 𝐵 ∈ [ 𝑥 ] 𝑅 ) )
2	1	expr	⊢ ( ( 𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [ 𝑥 ] 𝑅 ) → ( 𝐵 ∈ [ 𝑥 ] 𝑅 → ∃ 𝑥 ∈ dom 𝑅 ( 𝐴 ∈ [ 𝑥 ] 𝑅 ∧ 𝐵 ∈ [ 𝑥 ] 𝑅 ) ) )
3	2	adantl	⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) ∧ Disj 𝑅 ) ∧ ( 𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [ 𝑥 ] 𝑅 ) ) → ( 𝐵 ∈ [ 𝑥 ] 𝑅 → ∃ 𝑥 ∈ dom 𝑅 ( 𝐴 ∈ [ 𝑥 ] 𝑅 ∧ 𝐵 ∈ [ 𝑥 ] 𝑅 ) ) )
4		relbrcoss	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( Rel 𝑅 → ( 𝐴 ≀ 𝑅 𝐵 ↔ ∃ 𝑥 ∈ dom 𝑅 ( 𝐴 ∈ [ 𝑥 ] 𝑅 ∧ 𝐵 ∈ [ 𝑥 ] 𝑅 ) ) ) )
5		disjrel	⊢ ( Disj 𝑅 → Rel 𝑅 )
6	4 5	impel	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) ∧ Disj 𝑅 ) → ( 𝐴 ≀ 𝑅 𝐵 ↔ ∃ 𝑥 ∈ dom 𝑅 ( 𝐴 ∈ [ 𝑥 ] 𝑅 ∧ 𝐵 ∈ [ 𝑥 ] 𝑅 ) ) )
7	6	adantr	⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) ∧ Disj 𝑅 ) ∧ ( 𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [ 𝑥 ] 𝑅 ) ) → ( 𝐴 ≀ 𝑅 𝐵 ↔ ∃ 𝑥 ∈ dom 𝑅 ( 𝐴 ∈ [ 𝑥 ] 𝑅 ∧ 𝐵 ∈ [ 𝑥 ] 𝑅 ) ) )
8	3 7	sylibrd	⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) ∧ Disj 𝑅 ) ∧ ( 𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [ 𝑥 ] 𝑅 ) ) → ( 𝐵 ∈ [ 𝑥 ] 𝑅 → 𝐴 ≀ 𝑅 𝐵 ) )
9	8	ex	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) ∧ Disj 𝑅 ) → ( ( 𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [ 𝑥 ] 𝑅 ) → ( 𝐵 ∈ [ 𝑥 ] 𝑅 → 𝐴 ≀ 𝑅 𝐵 ) ) )
10		disjlem17	⊢ ( Disj 𝑅 → ( ( 𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [ 𝑥 ] 𝑅 ) → ( ∃ 𝑦 ∈ dom 𝑅 ( 𝐴 ∈ [ 𝑦 ] 𝑅 ∧ 𝐵 ∈ [ 𝑦 ] 𝑅 ) → 𝐵 ∈ [ 𝑥 ] 𝑅 ) ) )
11	10	adantl	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) ∧ Disj 𝑅 ) → ( ( 𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [ 𝑥 ] 𝑅 ) → ( ∃ 𝑦 ∈ dom 𝑅 ( 𝐴 ∈ [ 𝑦 ] 𝑅 ∧ 𝐵 ∈ [ 𝑦 ] 𝑅 ) → 𝐵 ∈ [ 𝑥 ] 𝑅 ) ) )
12		relbrcoss	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( Rel 𝑅 → ( 𝐴 ≀ 𝑅 𝐵 ↔ ∃ 𝑦 ∈ dom 𝑅 ( 𝐴 ∈ [ 𝑦 ] 𝑅 ∧ 𝐵 ∈ [ 𝑦 ] 𝑅 ) ) ) )
13	12 5	impel	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) ∧ Disj 𝑅 ) → ( 𝐴 ≀ 𝑅 𝐵 ↔ ∃ 𝑦 ∈ dom 𝑅 ( 𝐴 ∈ [ 𝑦 ] 𝑅 ∧ 𝐵 ∈ [ 𝑦 ] 𝑅 ) ) )
14	13	imbi1d	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) ∧ Disj 𝑅 ) → ( ( 𝐴 ≀ 𝑅 𝐵 → 𝐵 ∈ [ 𝑥 ] 𝑅 ) ↔ ( ∃ 𝑦 ∈ dom 𝑅 ( 𝐴 ∈ [ 𝑦 ] 𝑅 ∧ 𝐵 ∈ [ 𝑦 ] 𝑅 ) → 𝐵 ∈ [ 𝑥 ] 𝑅 ) ) )
15	11 14	sylibrd	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) ∧ Disj 𝑅 ) → ( ( 𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [ 𝑥 ] 𝑅 ) → ( 𝐴 ≀ 𝑅 𝐵 → 𝐵 ∈ [ 𝑥 ] 𝑅 ) ) )
16	9 15	impbidd	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) ∧ Disj 𝑅 ) → ( ( 𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [ 𝑥 ] 𝑅 ) → ( 𝐵 ∈ [ 𝑥 ] 𝑅 ↔ 𝐴 ≀ 𝑅 𝐵 ) ) )
17	16	ex	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( Disj 𝑅 → ( ( 𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [ 𝑥 ] 𝑅 ) → ( 𝐵 ∈ [ 𝑥 ] 𝑅 ↔ 𝐴 ≀ 𝑅 𝐵 ) ) ) )