Theorem imaiun

Description: The image of an indexed union is the indexed union of the images. (Contributed by Mario Carneiro, 18-Jun-2014)

	Ref	Expression
Assertion	imaiun	⊢ ( 𝐴 “ ∪ 𝑥 ∈ 𝐵 𝐶 ) = ∪ 𝑥 ∈ 𝐵 ( 𝐴 “ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		rexcom4	⊢ ( ∃ 𝑥 ∈ 𝐵 ∃ 𝑧 ( 𝑧 ∈ 𝐶 ∧ ⟨ 𝑧 , 𝑦 ⟩ ∈ 𝐴 ) ↔ ∃ 𝑧 ∃ 𝑥 ∈ 𝐵 ( 𝑧 ∈ 𝐶 ∧ ⟨ 𝑧 , 𝑦 ⟩ ∈ 𝐴 ) )
2		vex	⊢ 𝑦 ∈ V
3	2	elima3	⊢ ( 𝑦 ∈ ( 𝐴 “ 𝐶 ) ↔ ∃ 𝑧 ( 𝑧 ∈ 𝐶 ∧ ⟨ 𝑧 , 𝑦 ⟩ ∈ 𝐴 ) )
4	3	rexbii	⊢ ( ∃ 𝑥 ∈ 𝐵 𝑦 ∈ ( 𝐴 “ 𝐶 ) ↔ ∃ 𝑥 ∈ 𝐵 ∃ 𝑧 ( 𝑧 ∈ 𝐶 ∧ ⟨ 𝑧 , 𝑦 ⟩ ∈ 𝐴 ) )
5		eliun	⊢ ( 𝑧 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∃ 𝑥 ∈ 𝐵 𝑧 ∈ 𝐶 )
6	5	anbi1i	⊢ ( ( 𝑧 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ∧ ⟨ 𝑧 , 𝑦 ⟩ ∈ 𝐴 ) ↔ ( ∃ 𝑥 ∈ 𝐵 𝑧 ∈ 𝐶 ∧ ⟨ 𝑧 , 𝑦 ⟩ ∈ 𝐴 ) )
7		r19.41v	⊢ ( ∃ 𝑥 ∈ 𝐵 ( 𝑧 ∈ 𝐶 ∧ ⟨ 𝑧 , 𝑦 ⟩ ∈ 𝐴 ) ↔ ( ∃ 𝑥 ∈ 𝐵 𝑧 ∈ 𝐶 ∧ ⟨ 𝑧 , 𝑦 ⟩ ∈ 𝐴 ) )
8	6 7	bitr4i	⊢ ( ( 𝑧 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ∧ ⟨ 𝑧 , 𝑦 ⟩ ∈ 𝐴 ) ↔ ∃ 𝑥 ∈ 𝐵 ( 𝑧 ∈ 𝐶 ∧ ⟨ 𝑧 , 𝑦 ⟩ ∈ 𝐴 ) )
9	8	exbii	⊢ ( ∃ 𝑧 ( 𝑧 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ∧ ⟨ 𝑧 , 𝑦 ⟩ ∈ 𝐴 ) ↔ ∃ 𝑧 ∃ 𝑥 ∈ 𝐵 ( 𝑧 ∈ 𝐶 ∧ ⟨ 𝑧 , 𝑦 ⟩ ∈ 𝐴 ) )
10	1 4 9	3bitr4ri	⊢ ( ∃ 𝑧 ( 𝑧 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ∧ ⟨ 𝑧 , 𝑦 ⟩ ∈ 𝐴 ) ↔ ∃ 𝑥 ∈ 𝐵 𝑦 ∈ ( 𝐴 “ 𝐶 ) )
11	2	elima3	⊢ ( 𝑦 ∈ ( 𝐴 “ ∪ 𝑥 ∈ 𝐵 𝐶 ) ↔ ∃ 𝑧 ( 𝑧 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ∧ ⟨ 𝑧 , 𝑦 ⟩ ∈ 𝐴 ) )
12		eliun	⊢ ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐵 ( 𝐴 “ 𝐶 ) ↔ ∃ 𝑥 ∈ 𝐵 𝑦 ∈ ( 𝐴 “ 𝐶 ) )
13	10 11 12	3bitr4i	⊢ ( 𝑦 ∈ ( 𝐴 “ ∪ 𝑥 ∈ 𝐵 𝐶 ) ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐵 ( 𝐴 “ 𝐶 ) )
14	13	eqriv	⊢ ( 𝐴 “ ∪ 𝑥 ∈ 𝐵 𝐶 ) = ∪ 𝑥 ∈ 𝐵 ( 𝐴 “ 𝐶 )