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Metamath Proof Explorer

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