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Metamath Proof Explorer

Theorem imaco

Description: Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006) (Proof shortened by Wolf Lammen, 16-May-2025)

Ref Expression
Assertion imaco ( ( 𝐴 ∘ 𝐵 ) “ 𝐶 ) = ( 𝐴 “ ( 𝐵 “ 𝐶 ) )

Proof

Step Hyp Ref Expression
1 df-rex ⊢ ( ∃ 𝑦 ∈ ( 𝐵 “ 𝐶 ) 𝑦 𝐴 𝑥 ↔ ∃ 𝑦 ( 𝑦 ∈ ( 𝐵 “ 𝐶 ) ∧ 𝑦 𝐴 𝑥 ) )
2 vex ⊢ 𝑥 ∈ V
3 2 elima ⊢ ( 𝑥 ∈ ( 𝐴 “ ( 𝐵 “ 𝐶 ) ) ↔ ∃ 𝑦 ∈ ( 𝐵 “ 𝐶 ) 𝑦 𝐴 𝑥 )
4 vex ⊢ 𝑧 ∈ V
5 4 2 brco ⊢ ( 𝑧 ( 𝐴 ∘ 𝐵 ) 𝑥 ↔ ∃ 𝑦 ( 𝑧 𝐵 𝑦 ∧ 𝑦 𝐴 𝑥 ) )
6 5 rexbii ⊢ ( ∃ 𝑧 ∈ 𝐶 𝑧 ( 𝐴 ∘ 𝐵 ) 𝑥 ↔ ∃ 𝑧 ∈ 𝐶 ∃ 𝑦 ( 𝑧 𝐵 𝑦 ∧ 𝑦 𝐴 𝑥 ) )
7 rexcom4 ⊢ ( ∃ 𝑧 ∈ 𝐶 ∃ 𝑦 ( 𝑧 𝐵 𝑦 ∧ 𝑦 𝐴 𝑥 ) ↔ ∃ 𝑦 ∃ 𝑧 ∈ 𝐶 ( 𝑧 𝐵 𝑦 ∧ 𝑦 𝐴 𝑥 ) )
8 r19.41v ⊢ ( ∃ 𝑧 ∈ 𝐶 ( 𝑧 𝐵 𝑦 ∧ 𝑦 𝐴 𝑥 ) ↔ ( ∃ 𝑧 ∈ 𝐶 𝑧 𝐵 𝑦 ∧ 𝑦 𝐴 𝑥 ) )
9 8 exbii ⊢ ( ∃ 𝑦 ∃ 𝑧 ∈ 𝐶 ( 𝑧 𝐵 𝑦 ∧ 𝑦 𝐴 𝑥 ) ↔ ∃ 𝑦 ( ∃ 𝑧 ∈ 𝐶 𝑧 𝐵 𝑦 ∧ 𝑦 𝐴 𝑥 ) )
10 6 7 9 3bitri ⊢ ( ∃ 𝑧 ∈ 𝐶 𝑧 ( 𝐴 ∘ 𝐵 ) 𝑥 ↔ ∃ 𝑦 ( ∃ 𝑧 ∈ 𝐶 𝑧 𝐵 𝑦 ∧ 𝑦 𝐴 𝑥 ) )
11 2 elima ⊢ ( 𝑥 ∈ ( ( 𝐴 ∘ 𝐵 ) “ 𝐶 ) ↔ ∃ 𝑧 ∈ 𝐶 𝑧 ( 𝐴 ∘ 𝐵 ) 𝑥 )
12 vex ⊢ 𝑦 ∈ V
13 12 elima ⊢ ( 𝑦 ∈ ( 𝐵 “ 𝐶 ) ↔ ∃ 𝑧 ∈ 𝐶 𝑧 𝐵 𝑦 )
14 13 anbi1i ⊢ ( ( 𝑦 ∈ ( 𝐵 “ 𝐶 ) ∧ 𝑦 𝐴 𝑥 ) ↔ ( ∃ 𝑧 ∈ 𝐶 𝑧 𝐵 𝑦 ∧ 𝑦 𝐴 𝑥 ) )
15 14 exbii ⊢ ( ∃ 𝑦 ( 𝑦 ∈ ( 𝐵 “ 𝐶 ) ∧ 𝑦 𝐴 𝑥 ) ↔ ∃ 𝑦 ( ∃ 𝑧 ∈ 𝐶 𝑧 𝐵 𝑦 ∧ 𝑦 𝐴 𝑥 ) )
16 10 11 15 3bitr4i ⊢ ( 𝑥 ∈ ( ( 𝐴 ∘ 𝐵 ) “ 𝐶 ) ↔ ∃ 𝑦 ( 𝑦 ∈ ( 𝐵 “ 𝐶 ) ∧ 𝑦 𝐴 𝑥 ) )
17 1 3 16 3bitr4ri ⊢ ( 𝑥 ∈ ( ( 𝐴 ∘ 𝐵 ) “ 𝐶 ) ↔ 𝑥 ∈ ( 𝐴 “ ( 𝐵 “ 𝐶 ) ) )
18 17 eqriv ⊢ ( ( 𝐴 ∘ 𝐵 ) “ 𝐶 ) = ( 𝐴 “ ( 𝐵 “ 𝐶 ) )