Theorem imaiun1

Description: The image of an indexed union is the indexed union of the images. (Contributed by RP, 29-Jun-2020)

	Ref	Expression
Assertion	imaiun1	⊢ ( ∪ 𝑥 ∈ 𝐴 𝐵 “ 𝐶 ) = ∪ 𝑥 ∈ 𝐴 ( 𝐵 “ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		rexcom4	⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑧 ( 𝑧 ∈ 𝐶 ∧ ⟨ 𝑧 , 𝑦 ⟩ ∈ 𝐵 ) ↔ ∃ 𝑧 ∃ 𝑥 ∈ 𝐴 ( 𝑧 ∈ 𝐶 ∧ ⟨ 𝑧 , 𝑦 ⟩ ∈ 𝐵 ) )
2		vex	⊢ 𝑦 ∈ V
3	2	elima3	⊢ ( 𝑦 ∈ ( 𝐵 “ 𝐶 ) ↔ ∃ 𝑧 ( 𝑧 ∈ 𝐶 ∧ ⟨ 𝑧 , 𝑦 ⟩ ∈ 𝐵 ) )
4	3	rexbii	⊢ ( ∃ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 “ 𝐶 ) ↔ ∃ 𝑥 ∈ 𝐴 ∃ 𝑧 ( 𝑧 ∈ 𝐶 ∧ ⟨ 𝑧 , 𝑦 ⟩ ∈ 𝐵 ) )
5		eliun	⊢ ( ⟨ 𝑧 , 𝑦 ⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃ 𝑥 ∈ 𝐴 ⟨ 𝑧 , 𝑦 ⟩ ∈ 𝐵 )
6	5	anbi2i	⊢ ( ( 𝑧 ∈ 𝐶 ∧ ⟨ 𝑧 , 𝑦 ⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ↔ ( 𝑧 ∈ 𝐶 ∧ ∃ 𝑥 ∈ 𝐴 ⟨ 𝑧 , 𝑦 ⟩ ∈ 𝐵 ) )
7		r19.42v	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝑧 ∈ 𝐶 ∧ ⟨ 𝑧 , 𝑦 ⟩ ∈ 𝐵 ) ↔ ( 𝑧 ∈ 𝐶 ∧ ∃ 𝑥 ∈ 𝐴 ⟨ 𝑧 , 𝑦 ⟩ ∈ 𝐵 ) )
8	6 7	bitr4i	⊢ ( ( 𝑧 ∈ 𝐶 ∧ ⟨ 𝑧 , 𝑦 ⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ↔ ∃ 𝑥 ∈ 𝐴 ( 𝑧 ∈ 𝐶 ∧ ⟨ 𝑧 , 𝑦 ⟩ ∈ 𝐵 ) )
9	8	exbii	⊢ ( ∃ 𝑧 ( 𝑧 ∈ 𝐶 ∧ ⟨ 𝑧 , 𝑦 ⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ↔ ∃ 𝑧 ∃ 𝑥 ∈ 𝐴 ( 𝑧 ∈ 𝐶 ∧ ⟨ 𝑧 , 𝑦 ⟩ ∈ 𝐵 ) )
10	1 4 9	3bitr4ri	⊢ ( ∃ 𝑧 ( 𝑧 ∈ 𝐶 ∧ ⟨ 𝑧 , 𝑦 ⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ↔ ∃ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 “ 𝐶 ) )
11	2	elima3	⊢ ( 𝑦 ∈ ( ∪ 𝑥 ∈ 𝐴 𝐵 “ 𝐶 ) ↔ ∃ 𝑧 ( 𝑧 ∈ 𝐶 ∧ ⟨ 𝑧 , 𝑦 ⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) )
12		eliun	⊢ ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ( 𝐵 “ 𝐶 ) ↔ ∃ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 “ 𝐶 ) )
13	10 11 12	3bitr4i	⊢ ( 𝑦 ∈ ( ∪ 𝑥 ∈ 𝐴 𝐵 “ 𝐶 ) ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ( 𝐵 “ 𝐶 ) )
14	13	eqriv	⊢ ( ∪ 𝑥 ∈ 𝐴 𝐵 “ 𝐶 ) = ∪ 𝑥 ∈ 𝐴 ( 𝐵 “ 𝐶 )