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Metamath Proof Explorer

Theorem unima

Description: Image of a union. (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Assertion unima ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴 ) → ( 𝐹 “ ( 𝐵 ∪ 𝐶 ) ) = ( ( 𝐹 “ 𝐵 ) ∪ ( 𝐹 “ 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 simp1 ⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴 ) → 𝐹 Fn 𝐴 )
2 simpl ⊢ ( ( 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴 ) → 𝐵 ⊆ 𝐴 )
3 simpr ⊢ ( ( 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴 ) → 𝐶 ⊆ 𝐴 )
4 2 3 unssd ⊢ ( ( 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴 ) → ( 𝐵 ∪ 𝐶 ) ⊆ 𝐴 )
5 4 3adant1 ⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴 ) → ( 𝐵 ∪ 𝐶 ) ⊆ 𝐴 )
6 1 5 fvelimabd ⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴 ) → ( 𝑦 ∈ ( 𝐹 “ ( 𝐵 ∪ 𝐶 ) ) ↔ ∃ 𝑥 ∈ ( 𝐵 ∪ 𝐶 ) ( 𝐹 ‘ 𝑥 ) = 𝑦 ) )
7 rexun ⊢ ( ∃ 𝑥 ∈ ( 𝐵 ∪ 𝐶 ) ( 𝐹 ‘ 𝑥 ) = 𝑦 ↔ ( ∃ 𝑥 ∈ 𝐵 ( 𝐹 ‘ 𝑥 ) = 𝑦 ∨ ∃ 𝑥 ∈ 𝐶 ( 𝐹 ‘ 𝑥 ) = 𝑦 ) )
8 6 7 bitrdi ⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴 ) → ( 𝑦 ∈ ( 𝐹 “ ( 𝐵 ∪ 𝐶 ) ) ↔ ( ∃ 𝑥 ∈ 𝐵 ( 𝐹 ‘ 𝑥 ) = 𝑦 ∨ ∃ 𝑥 ∈ 𝐶 ( 𝐹 ‘ 𝑥 ) = 𝑦 ) ) )
9 fvelimab ⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ) → ( 𝑦 ∈ ( 𝐹 “ 𝐵 ) ↔ ∃ 𝑥 ∈ 𝐵 ( 𝐹 ‘ 𝑥 ) = 𝑦 ) )
10 9 3adant3 ⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴 ) → ( 𝑦 ∈ ( 𝐹 “ 𝐵 ) ↔ ∃ 𝑥 ∈ 𝐵 ( 𝐹 ‘ 𝑥 ) = 𝑦 ) )
11 fvelimab ⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐶 ⊆ 𝐴 ) → ( 𝑦 ∈ ( 𝐹 “ 𝐶 ) ↔ ∃ 𝑥 ∈ 𝐶 ( 𝐹 ‘ 𝑥 ) = 𝑦 ) )
12 11 3adant2 ⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴 ) → ( 𝑦 ∈ ( 𝐹 “ 𝐶 ) ↔ ∃ 𝑥 ∈ 𝐶 ( 𝐹 ‘ 𝑥 ) = 𝑦 ) )
13 10 12 orbi12d ⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴 ) → ( ( 𝑦 ∈ ( 𝐹 “ 𝐵 ) ∨ 𝑦 ∈ ( 𝐹 “ 𝐶 ) ) ↔ ( ∃ 𝑥 ∈ 𝐵 ( 𝐹 ‘ 𝑥 ) = 𝑦 ∨ ∃ 𝑥 ∈ 𝐶 ( 𝐹 ‘ 𝑥 ) = 𝑦 ) ) )
14 8 13 bitr4d ⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴 ) → ( 𝑦 ∈ ( 𝐹 “ ( 𝐵 ∪ 𝐶 ) ) ↔ ( 𝑦 ∈ ( 𝐹 “ 𝐵 ) ∨ 𝑦 ∈ ( 𝐹 “ 𝐶 ) ) ) )
15 elun ⊢ ( 𝑦 ∈ ( ( 𝐹 “ 𝐵 ) ∪ ( 𝐹 “ 𝐶 ) ) ↔ ( 𝑦 ∈ ( 𝐹 “ 𝐵 ) ∨ 𝑦 ∈ ( 𝐹 “ 𝐶 ) ) )
16 14 15 bitr4di ⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴 ) → ( 𝑦 ∈ ( 𝐹 “ ( 𝐵 ∪ 𝐶 ) ) ↔ 𝑦 ∈ ( ( 𝐹 “ 𝐵 ) ∪ ( 𝐹 “ 𝐶 ) ) ) )
17 16 eqrdv ⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴 ) → ( 𝐹 “ ( 𝐵 ∪ 𝐶 ) ) = ( ( 𝐹 “ 𝐵 ) ∪ ( 𝐹 “ 𝐶 ) ) )