Metamath Proof Explorer

Theorem bj-imdirval3

Description: Value of the functionalized direct image. (Contributed by BJ, 16-Dec-2023)

Ref Expression
Hypotheses bj-imdirval3.exa ⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
bj-imdirval3.exb ⊢ ( 𝜑 → 𝐵 ∈ 𝑉 )
bj-imdirval3.arg ⊢ ( 𝜑 → 𝑅 ⊆ ( 𝐴 × 𝐵 ) )
Assertion bj-imdirval3 ( 𝜑 → ( 𝑋 ( ( 𝐴 𝒫* 𝐵 ) ‘ 𝑅 ) 𝑌 ↔ ( ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵 ) ∧ ( 𝑅 “ 𝑋 ) = 𝑌 ) ) )

Proof

Step Hyp Ref Expression
1 bj-imdirval3.exa ⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
2 bj-imdirval3.exb ⊢ ( 𝜑 → 𝐵 ∈ 𝑉 )
3 bj-imdirval3.arg ⊢ ( 𝜑 → 𝑅 ⊆ ( 𝐴 × 𝐵 ) )
4 1 2 3 bj-imdirval2 ⊢ ( 𝜑 → ( ( 𝐴 𝒫* 𝐵 ) ‘ 𝑅 ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵 ) ∧ ( 𝑅 “ 𝑥 ) = 𝑦 ) } )
5 4 breqd ⊢ ( 𝜑 → ( 𝑋 ( ( 𝐴 𝒫* 𝐵 ) ‘ 𝑅 ) 𝑌 ↔ 𝑋 { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵 ) ∧ ( 𝑅 “ 𝑥 ) = 𝑦 ) } 𝑌 ) )
6 brabv ⊢ ( 𝑋 { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵 ) ∧ ( 𝑅 “ 𝑥 ) = 𝑦 ) } 𝑌 → ( 𝑋 ∈ V ∧ 𝑌 ∈ V ) )
7 5 6 syl6bi ⊢ ( 𝜑 → ( 𝑋 ( ( 𝐴 𝒫* 𝐵 ) ‘ 𝑅 ) 𝑌 → ( 𝑋 ∈ V ∧ 𝑌 ∈ V ) ) )
8 7 pm4.71rd ⊢ ( 𝜑 → ( 𝑋 ( ( 𝐴 𝒫* 𝐵 ) ‘ 𝑅 ) 𝑌 ↔ ( ( 𝑋 ∈ V ∧ 𝑌 ∈ V ) ∧ 𝑋 ( ( 𝐴 𝒫* 𝐵 ) ‘ 𝑅 ) 𝑌 ) ) )
9 simpl ⊢ ( ( 𝑋 ∈ V ∧ 𝑌 ∈ V ) → 𝑋 ∈ V )
10 9 adantl ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ V ∧ 𝑌 ∈ V ) ) → 𝑋 ∈ V )
11 simpr ⊢ ( ( 𝑋 ∈ V ∧ 𝑌 ∈ V ) → 𝑌 ∈ V )
12 11 adantl ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ V ∧ 𝑌 ∈ V ) ) → 𝑌 ∈ V )
13 4 adantr ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ V ∧ 𝑌 ∈ V ) ) → ( ( 𝐴 𝒫* 𝐵 ) ‘ 𝑅 ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵 ) ∧ ( 𝑅 “ 𝑥 ) = 𝑦 ) } )
14 simpl ⊢ ( ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) → 𝑥 = 𝑋 )
15 14 sseq1d ⊢ ( ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) → ( 𝑥 ⊆ 𝐴 ↔ 𝑋 ⊆ 𝐴 ) )
16 simpr ⊢ ( ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) → 𝑦 = 𝑌 )
17 16 sseq1d ⊢ ( ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) → ( 𝑦 ⊆ 𝐵 ↔ 𝑌 ⊆ 𝐵 ) )
18 15 17 anbi12d ⊢ ( ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) → ( ( 𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵 ) ↔ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵 ) ) )
19 imaeq2 ⊢ ( 𝑥 = 𝑋 → ( 𝑅 “ 𝑥 ) = ( 𝑅 “ 𝑋 ) )
20 id ⊢ ( 𝑦 = 𝑌 → 𝑦 = 𝑌 )
21 19 20 eqeqan12d ⊢ ( ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) → ( ( 𝑅 “ 𝑥 ) = 𝑦 ↔ ( 𝑅 “ 𝑋 ) = 𝑌 ) )
22 18 21 anbi12d ⊢ ( ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) → ( ( ( 𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵 ) ∧ ( 𝑅 “ 𝑥 ) = 𝑦 ) ↔ ( ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵 ) ∧ ( 𝑅 “ 𝑋 ) = 𝑌 ) ) )
23 22 adantl ⊢ ( ( ( 𝜑 ∧ ( 𝑋 ∈ V ∧ 𝑌 ∈ V ) ) ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( ( ( 𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵 ) ∧ ( 𝑅 “ 𝑥 ) = 𝑦 ) ↔ ( ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵 ) ∧ ( 𝑅 “ 𝑋 ) = 𝑌 ) ) )
24 10 12 13 23 brabd ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ V ∧ 𝑌 ∈ V ) ) → ( 𝑋 ( ( 𝐴 𝒫* 𝐵 ) ‘ 𝑅 ) 𝑌 ↔ ( ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵 ) ∧ ( 𝑅 “ 𝑋 ) = 𝑌 ) ) )
25 24 pm5.32da ⊢ ( 𝜑 → ( ( ( 𝑋 ∈ V ∧ 𝑌 ∈ V ) ∧ 𝑋 ( ( 𝐴 𝒫* 𝐵 ) ‘ 𝑅 ) 𝑌 ) ↔ ( ( 𝑋 ∈ V ∧ 𝑌 ∈ V ) ∧ ( ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵 ) ∧ ( 𝑅 “ 𝑋 ) = 𝑌 ) ) ) )
26 simpr ⊢ ( ( ( 𝑋 ∈ V ∧ 𝑌 ∈ V ) ∧ ( ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵 ) ∧ ( 𝑅 “ 𝑋 ) = 𝑌 ) ) → ( ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵 ) ∧ ( 𝑅 “ 𝑋 ) = 𝑌 ) )
27 1 adantr ⊢ ( ( 𝜑 ∧ 𝑋 ⊆ 𝐴 ) → 𝐴 ∈ 𝑈 )
28 simpr ⊢ ( ( 𝜑 ∧ 𝑋 ⊆ 𝐴 ) → 𝑋 ⊆ 𝐴 )
29 27 28 ssexd ⊢ ( ( 𝜑 ∧ 𝑋 ⊆ 𝐴 ) → 𝑋 ∈ V )
30 29 ex ⊢ ( 𝜑 → ( 𝑋 ⊆ 𝐴 → 𝑋 ∈ V ) )
31 2 adantr ⊢ ( ( 𝜑 ∧ 𝑌 ⊆ 𝐵 ) → 𝐵 ∈ 𝑉 )
32 simpr ⊢ ( ( 𝜑 ∧ 𝑌 ⊆ 𝐵 ) → 𝑌 ⊆ 𝐵 )
33 31 32 ssexd ⊢ ( ( 𝜑 ∧ 𝑌 ⊆ 𝐵 ) → 𝑌 ∈ V )
34 33 ex ⊢ ( 𝜑 → ( 𝑌 ⊆ 𝐵 → 𝑌 ∈ V ) )
35 30 34 anim12d ⊢ ( 𝜑 → ( ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵 ) → ( 𝑋 ∈ V ∧ 𝑌 ∈ V ) ) )
36 35 adantrd ⊢ ( 𝜑 → ( ( ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵 ) ∧ ( 𝑅 “ 𝑋 ) = 𝑌 ) → ( 𝑋 ∈ V ∧ 𝑌 ∈ V ) ) )
37 36 ancrd ⊢ ( 𝜑 → ( ( ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵 ) ∧ ( 𝑅 “ 𝑋 ) = 𝑌 ) → ( ( 𝑋 ∈ V ∧ 𝑌 ∈ V ) ∧ ( ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵 ) ∧ ( 𝑅 “ 𝑋 ) = 𝑌 ) ) ) )
38 26 37 impbid2 ⊢ ( 𝜑 → ( ( ( 𝑋 ∈ V ∧ 𝑌 ∈ V ) ∧ ( ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵 ) ∧ ( 𝑅 “ 𝑋 ) = 𝑌 ) ) ↔ ( ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵 ) ∧ ( 𝑅 “ 𝑋 ) = 𝑌 ) ) )
39 8 25 38 3bitrd ⊢ ( 𝜑 → ( 𝑋 ( ( 𝐴 𝒫* 𝐵 ) ‘ 𝑅 ) 𝑌 ↔ ( ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵 ) ∧ ( 𝑅 “ 𝑋 ) = 𝑌 ) ) )