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Metamath Proof Explorer

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 ) ) ) )