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Metamath Proof Explorer

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 )