Database
SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
Mathbox for Alexander van der Vekens
Basic algebraic structures (extension)
Auxiliary theorems
ovmpordx
Metamath Proof Explorer
Description: Value of an operation given by a maps-to rule, deduction form, with
substitution of second argument, analogous to ovmpodxf . (Contributed by AV , 30-Mar-2019)
Ref
Expression
Hypotheses
ovmpordx.1
⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐶 , 𝑦 ∈ 𝐷 ↦ 𝑅 ) )
ovmpordx.2
⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ) → 𝑅 = 𝑆 )
ovmpordx.3
⊢ ( ( 𝜑 ∧ 𝑦 = 𝐵 ) → 𝐶 = 𝐿 )
ovmpordx.4
⊢ ( 𝜑 → 𝐴 ∈ 𝐿 )
ovmpordx.5
⊢ ( 𝜑 → 𝐵 ∈ 𝐷 )
ovmpordx.6
⊢ ( 𝜑 → 𝑆 ∈ 𝑋 )
Assertion
ovmpordx
⊢ ( 𝜑 → ( 𝐴 𝐹 𝐵 ) = 𝑆 )
Proof
Step
Hyp
Ref
Expression
1
ovmpordx.1
⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐶 , 𝑦 ∈ 𝐷 ↦ 𝑅 ) )
2
ovmpordx.2
⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ) → 𝑅 = 𝑆 )
3
ovmpordx.3
⊢ ( ( 𝜑 ∧ 𝑦 = 𝐵 ) → 𝐶 = 𝐿 )
4
ovmpordx.4
⊢ ( 𝜑 → 𝐴 ∈ 𝐿 )
5
ovmpordx.5
⊢ ( 𝜑 → 𝐵 ∈ 𝐷 )
6
ovmpordx.6
⊢ ( 𝜑 → 𝑆 ∈ 𝑋 )
7
nfv
⊢ Ⅎ 𝑥 𝜑
8
nfv
⊢ Ⅎ 𝑦 𝜑
9
nfcv
⊢ Ⅎ 𝑦 𝐴
10
nfcv
⊢ Ⅎ 𝑥 𝐵
11
nfcv
⊢ Ⅎ 𝑥 𝑆
12
nfcv
⊢ Ⅎ 𝑦 𝑆
13
1 2 3 4 5 6 7 8 9 10 11 12
ovmpordxf
⊢ ( 𝜑 → ( 𝐴 𝐹 𝐵 ) = 𝑆 )