Metamath Proof Explorer


Theorem unima

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

Ref Expression
Assertion unima ( ( 𝐹 Fn 𝐴𝐵𝐴𝐶𝐴 ) → ( 𝐹 “ ( 𝐵𝐶 ) ) = ( ( 𝐹𝐵 ) ∪ ( 𝐹𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 simp1 ( ( 𝐹 Fn 𝐴𝐵𝐴𝐶𝐴 ) → 𝐹 Fn 𝐴 )
2 simpl ( ( 𝐵𝐴𝐶𝐴 ) → 𝐵𝐴 )
3 simpr ( ( 𝐵𝐴𝐶𝐴 ) → 𝐶𝐴 )
4 2 3 unssd ( ( 𝐵𝐴𝐶𝐴 ) → ( 𝐵𝐶 ) ⊆ 𝐴 )
5 4 3adant1 ( ( 𝐹 Fn 𝐴𝐵𝐴𝐶𝐴 ) → ( 𝐵𝐶 ) ⊆ 𝐴 )
6 1 5 fvelimabd ( ( 𝐹 Fn 𝐴𝐵𝐴𝐶𝐴 ) → ( 𝑦 ∈ ( 𝐹 “ ( 𝐵𝐶 ) ) ↔ ∃ 𝑥 ∈ ( 𝐵𝐶 ) ( 𝐹𝑥 ) = 𝑦 ) )
7 rexun ( ∃ 𝑥 ∈ ( 𝐵𝐶 ) ( 𝐹𝑥 ) = 𝑦 ↔ ( ∃ 𝑥𝐵 ( 𝐹𝑥 ) = 𝑦 ∨ ∃ 𝑥𝐶 ( 𝐹𝑥 ) = 𝑦 ) )
8 6 7 bitrdi ( ( 𝐹 Fn 𝐴𝐵𝐴𝐶𝐴 ) → ( 𝑦 ∈ ( 𝐹 “ ( 𝐵𝐶 ) ) ↔ ( ∃ 𝑥𝐵 ( 𝐹𝑥 ) = 𝑦 ∨ ∃ 𝑥𝐶 ( 𝐹𝑥 ) = 𝑦 ) ) )
9 fvelimab ( ( 𝐹 Fn 𝐴𝐵𝐴 ) → ( 𝑦 ∈ ( 𝐹𝐵 ) ↔ ∃ 𝑥𝐵 ( 𝐹𝑥 ) = 𝑦 ) )
10 9 3adant3 ( ( 𝐹 Fn 𝐴𝐵𝐴𝐶𝐴 ) → ( 𝑦 ∈ ( 𝐹𝐵 ) ↔ ∃ 𝑥𝐵 ( 𝐹𝑥 ) = 𝑦 ) )
11 fvelimab ( ( 𝐹 Fn 𝐴𝐶𝐴 ) → ( 𝑦 ∈ ( 𝐹𝐶 ) ↔ ∃ 𝑥𝐶 ( 𝐹𝑥 ) = 𝑦 ) )
12 11 3adant2 ( ( 𝐹 Fn 𝐴𝐵𝐴𝐶𝐴 ) → ( 𝑦 ∈ ( 𝐹𝐶 ) ↔ ∃ 𝑥𝐶 ( 𝐹𝑥 ) = 𝑦 ) )
13 10 12 orbi12d ( ( 𝐹 Fn 𝐴𝐵𝐴𝐶𝐴 ) → ( ( 𝑦 ∈ ( 𝐹𝐵 ) ∨ 𝑦 ∈ ( 𝐹𝐶 ) ) ↔ ( ∃ 𝑥𝐵 ( 𝐹𝑥 ) = 𝑦 ∨ ∃ 𝑥𝐶 ( 𝐹𝑥 ) = 𝑦 ) ) )
14 8 13 bitr4d ( ( 𝐹 Fn 𝐴𝐵𝐴𝐶𝐴 ) → ( 𝑦 ∈ ( 𝐹 “ ( 𝐵𝐶 ) ) ↔ ( 𝑦 ∈ ( 𝐹𝐵 ) ∨ 𝑦 ∈ ( 𝐹𝐶 ) ) ) )
15 elun ( 𝑦 ∈ ( ( 𝐹𝐵 ) ∪ ( 𝐹𝐶 ) ) ↔ ( 𝑦 ∈ ( 𝐹𝐵 ) ∨ 𝑦 ∈ ( 𝐹𝐶 ) ) )
16 14 15 bitr4di ( ( 𝐹 Fn 𝐴𝐵𝐴𝐶𝐴 ) → ( 𝑦 ∈ ( 𝐹 “ ( 𝐵𝐶 ) ) ↔ 𝑦 ∈ ( ( 𝐹𝐵 ) ∪ ( 𝐹𝐶 ) ) ) )
17 16 eqrdv ( ( 𝐹 Fn 𝐴𝐵𝐴𝐶𝐴 ) → ( 𝐹 “ ( 𝐵𝐶 ) ) = ( ( 𝐹𝐵 ) ∪ ( 𝐹𝐶 ) ) )