Theorem iunrdx

Description: Re-index an indexed union. (Contributed by Thierry Arnoux, 6-Apr-2017)

	Ref	Expression
Hypotheses	iunrdx.1	⊢ ( 𝜑 → 𝐹 : 𝐴 –onto→ 𝐶 )
	iunrdx.2	⊢ ( ( 𝜑 ∧ 𝑦 = ( 𝐹 ‘ 𝑥 ) ) → 𝐷 = 𝐵 )
Assertion	iunrdx	⊢ ( 𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐶 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		iunrdx.1	⊢ ( 𝜑 → 𝐹 : 𝐴 –onto→ 𝐶 )
2		iunrdx.2	⊢ ( ( 𝜑 ∧ 𝑦 = ( 𝐹 ‘ 𝑥 ) ) → 𝐷 = 𝐵 )
3		fof	⊢ ( 𝐹 : 𝐴 –onto→ 𝐶 → 𝐹 : 𝐴 ⟶ 𝐶 )
4	1 3	syl	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐶 )
5	4	ffvelcdmda	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑥 ) ∈ 𝐶 )
6		foelrn	⊢ ( ( 𝐹 : 𝐴 –onto→ 𝐶 ∧ 𝑦 ∈ 𝐶 ) → ∃ 𝑥 ∈ 𝐴 𝑦 = ( 𝐹 ‘ 𝑥 ) )
7	1 6	sylan	⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ) → ∃ 𝑥 ∈ 𝐴 𝑦 = ( 𝐹 ‘ 𝑥 ) )
8	2	eleq2d	⊢ ( ( 𝜑 ∧ 𝑦 = ( 𝐹 ‘ 𝑥 ) ) → ( 𝑧 ∈ 𝐷 ↔ 𝑧 ∈ 𝐵 ) )
9	5 7 8	rexxfrd	⊢ ( 𝜑 → ( ∃ 𝑦 ∈ 𝐶 𝑧 ∈ 𝐷 ↔ ∃ 𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ) )
10	9	bicomd	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃ 𝑦 ∈ 𝐶 𝑧 ∈ 𝐷 ) )
11	10	abbidv	⊢ ( 𝜑 → { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 } = { 𝑧 ∣ ∃ 𝑦 ∈ 𝐶 𝑧 ∈ 𝐷 } )
12		df-iun	⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 }
13		df-iun	⊢ ∪ 𝑦 ∈ 𝐶 𝐷 = { 𝑧 ∣ ∃ 𝑦 ∈ 𝐶 𝑧 ∈ 𝐷 }
14	11 12 13	3eqtr4g	⊢ ( 𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐶 𝐷 )