Theorem iunxpf

Description: Indexed union on a Cartesian product equals a double indexed union. The hypothesis specifies an implicit substitution. (Contributed by NM, 19-Dec-2008)

	Ref	Expression
Hypotheses	iunxpf.1	⊢ Ⅎ 𝑦 𝐶
	iunxpf.2	⊢ Ⅎ 𝑧 𝐶
	iunxpf.3	⊢ Ⅎ 𝑥 𝐷
	iunxpf.4	⊢ ( 𝑥 = ⟨ 𝑦 , 𝑧 ⟩ → 𝐶 = 𝐷 )
Assertion	iunxpf	⊢ ∪ 𝑥 ∈ ( 𝐴 × 𝐵 ) 𝐶 = ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷

Proof

Step	Hyp	Ref	Expression
1		iunxpf.1	⊢ Ⅎ 𝑦 𝐶
2		iunxpf.2	⊢ Ⅎ 𝑧 𝐶
3		iunxpf.3	⊢ Ⅎ 𝑥 𝐷
4		iunxpf.4	⊢ ( 𝑥 = ⟨ 𝑦 , 𝑧 ⟩ → 𝐶 = 𝐷 )
5	1	nfcri	⊢ Ⅎ 𝑦 𝑤 ∈ 𝐶
6	2	nfcri	⊢ Ⅎ 𝑧 𝑤 ∈ 𝐶
7	3	nfcri	⊢ Ⅎ 𝑥 𝑤 ∈ 𝐷
8	4	eleq2d	⊢ ( 𝑥 = ⟨ 𝑦 , 𝑧 ⟩ → ( 𝑤 ∈ 𝐶 ↔ 𝑤 ∈ 𝐷 ) )
9	5 6 7 8	rexxpf	⊢ ( ∃ 𝑥 ∈ ( 𝐴 × 𝐵 ) 𝑤 ∈ 𝐶 ↔ ∃ 𝑦 ∈ 𝐴 ∃ 𝑧 ∈ 𝐵 𝑤 ∈ 𝐷 )
10		eliun	⊢ ( 𝑤 ∈ ∪ 𝑥 ∈ ( 𝐴 × 𝐵 ) 𝐶 ↔ ∃ 𝑥 ∈ ( 𝐴 × 𝐵 ) 𝑤 ∈ 𝐶 )
11		eliun	⊢ ( 𝑤 ∈ ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷 ↔ ∃ 𝑦 ∈ 𝐴 𝑤 ∈ ∪ 𝑧 ∈ 𝐵 𝐷 )
12		eliun	⊢ ( 𝑤 ∈ ∪ 𝑧 ∈ 𝐵 𝐷 ↔ ∃ 𝑧 ∈ 𝐵 𝑤 ∈ 𝐷 )
13	12	rexbii	⊢ ( ∃ 𝑦 ∈ 𝐴 𝑤 ∈ ∪ 𝑧 ∈ 𝐵 𝐷 ↔ ∃ 𝑦 ∈ 𝐴 ∃ 𝑧 ∈ 𝐵 𝑤 ∈ 𝐷 )
14	11 13	bitri	⊢ ( 𝑤 ∈ ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷 ↔ ∃ 𝑦 ∈ 𝐴 ∃ 𝑧 ∈ 𝐵 𝑤 ∈ 𝐷 )
15	9 10 14	3bitr4i	⊢ ( 𝑤 ∈ ∪ 𝑥 ∈ ( 𝐴 × 𝐵 ) 𝐶 ↔ 𝑤 ∈ ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷 )
16	15	eqriv	⊢ ∪ 𝑥 ∈ ( 𝐴 × 𝐵 ) 𝐶 = ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷