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Metamath Proof Explorer

Theorem xrnresex

Description: Sufficient condition for a restricted range Cartesian product to be a set. (Contributed by Peter Mazsa, 16-Dec-2020) (Revised by Peter Mazsa, 7-Sep-2021)

Ref Expression
Assertion xrnresex ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ ( 𝑆 ↾ 𝐴 ) ∈ 𝑋 ) → ( 𝑅 ⋉ ( 𝑆 ↾ 𝐴 ) ) ∈ V )

Proof

Step Hyp Ref Expression
1 xrnres3 ⊢ ( ( 𝑅 ⋉ 𝑆 ) ↾ 𝐴 ) = ( ( 𝑅 ↾ 𝐴 ) ⋉ ( 𝑆 ↾ 𝐴 ) )
2 xrnres2 ⊢ ( ( 𝑅 ⋉ 𝑆 ) ↾ 𝐴 ) = ( 𝑅 ⋉ ( 𝑆 ↾ 𝐴 ) )
3 1 2 eqtr3i ⊢ ( ( 𝑅 ↾ 𝐴 ) ⋉ ( 𝑆 ↾ 𝐴 ) ) = ( 𝑅 ⋉ ( 𝑆 ↾ 𝐴 ) )
4 dfres4 ⊢ ( 𝑅 ↾ 𝐴 ) = ( 𝑅 ∩ ( 𝐴 × ran ( 𝑅 ↾ 𝐴 ) ) )
5 dfres4 ⊢ ( 𝑆 ↾ 𝐴 ) = ( 𝑆 ∩ ( 𝐴 × ran ( 𝑆 ↾ 𝐴 ) ) )
6 4 5 xrneq12i ⊢ ( ( 𝑅 ↾ 𝐴 ) ⋉ ( 𝑆 ↾ 𝐴 ) ) = ( ( 𝑅 ∩ ( 𝐴 × ran ( 𝑅 ↾ 𝐴 ) ) ) ⋉ ( 𝑆 ∩ ( 𝐴 × ran ( 𝑆 ↾ 𝐴 ) ) ) )
7 simp1 ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ ( 𝑆 ↾ 𝐴 ) ∈ 𝑋 ) → 𝐴 ∈ 𝑉 )
8 resexg ⊢ ( 𝑅 ∈ 𝑊 → ( 𝑅 ↾ 𝐴 ) ∈ V )
9 rnexg ⊢ ( ( 𝑅 ↾ 𝐴 ) ∈ V → ran ( 𝑅 ↾ 𝐴 ) ∈ V )
10 8 9 syl ⊢ ( 𝑅 ∈ 𝑊 → ran ( 𝑅 ↾ 𝐴 ) ∈ V )
11 10 3ad2ant2 ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ ( 𝑆 ↾ 𝐴 ) ∈ 𝑋 ) → ran ( 𝑅 ↾ 𝐴 ) ∈ V )
12 rnexg ⊢ ( ( 𝑆 ↾ 𝐴 ) ∈ 𝑋 → ran ( 𝑆 ↾ 𝐴 ) ∈ V )
13 12 3ad2ant3 ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ ( 𝑆 ↾ 𝐴 ) ∈ 𝑋 ) → ran ( 𝑆 ↾ 𝐴 ) ∈ V )
14 inxpxrn ⊢ ( ( 𝑅 ∩ ( 𝐴 × ran ( 𝑅 ↾ 𝐴 ) ) ) ⋉ ( 𝑆 ∩ ( 𝐴 × ran ( 𝑆 ↾ 𝐴 ) ) ) ) = ( ( 𝑅 ⋉ 𝑆 ) ∩ ( 𝐴 × ( ran ( 𝑅 ↾ 𝐴 ) × ran ( 𝑆 ↾ 𝐴 ) ) ) )
15 xrninxpex ⊢ ( ( 𝐴 ∈ 𝑉 ∧ ran ( 𝑅 ↾ 𝐴 ) ∈ V ∧ ran ( 𝑆 ↾ 𝐴 ) ∈ V ) → ( ( 𝑅 ⋉ 𝑆 ) ∩ ( 𝐴 × ( ran ( 𝑅 ↾ 𝐴 ) × ran ( 𝑆 ↾ 𝐴 ) ) ) ) ∈ V )
16 14 15 eqeltrid ⊢ ( ( 𝐴 ∈ 𝑉 ∧ ran ( 𝑅 ↾ 𝐴 ) ∈ V ∧ ran ( 𝑆 ↾ 𝐴 ) ∈ V ) → ( ( 𝑅 ∩ ( 𝐴 × ran ( 𝑅 ↾ 𝐴 ) ) ) ⋉ ( 𝑆 ∩ ( 𝐴 × ran ( 𝑆 ↾ 𝐴 ) ) ) ) ∈ V )
17 7 11 13 16 syl3anc ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ ( 𝑆 ↾ 𝐴 ) ∈ 𝑋 ) → ( ( 𝑅 ∩ ( 𝐴 × ran ( 𝑅 ↾ 𝐴 ) ) ) ⋉ ( 𝑆 ∩ ( 𝐴 × ran ( 𝑆 ↾ 𝐴 ) ) ) ) ∈ V )
18 6 17 eqeltrid ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ ( 𝑆 ↾ 𝐴 ) ∈ 𝑋 ) → ( ( 𝑅 ↾ 𝐴 ) ⋉ ( 𝑆 ↾ 𝐴 ) ) ∈ V )
19 3 18 eqeltrrid ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ ( 𝑆 ↾ 𝐴 ) ∈ 𝑋 ) → ( 𝑅 ⋉ ( 𝑆 ↾ 𝐴 ) ) ∈ V )