Metamath Proof Explorer


Theorem ov2ssiunov2

Description: Any particular operator value is the subset of the index union over a set of operator values. Generalized from rtrclreclem1 and rtrclreclem2 . (Contributed by RP, 4-Jun-2020)

Ref Expression
Hypothesis ov2ssiunov2.def 𝐶 = ( 𝑟 ∈ V ↦ 𝑛𝑁 ( 𝑟 𝑛 ) )
Assertion ov2ssiunov2 ( ( 𝑅𝑈𝑁𝑉𝑀𝑁 ) → ( 𝑅 𝑀 ) ⊆ ( 𝐶𝑅 ) )

Proof

Step Hyp Ref Expression
1 ov2ssiunov2.def 𝐶 = ( 𝑟 ∈ V ↦ 𝑛𝑁 ( 𝑟 𝑛 ) )
2 simp3 ( ( 𝑅𝑈𝑁𝑉𝑀𝑁 ) → 𝑀𝑁 )
3 simpr ( ( ( 𝑅𝑈𝑁𝑉𝑀𝑁 ) ∧ 𝑛 = 𝑀 ) → 𝑛 = 𝑀 )
4 3 oveq2d ( ( ( 𝑅𝑈𝑁𝑉𝑀𝑁 ) ∧ 𝑛 = 𝑀 ) → ( 𝑅 𝑛 ) = ( 𝑅 𝑀 ) )
5 4 eleq2d ( ( ( 𝑅𝑈𝑁𝑉𝑀𝑁 ) ∧ 𝑛 = 𝑀 ) → ( 𝑥 ∈ ( 𝑅 𝑛 ) ↔ 𝑥 ∈ ( 𝑅 𝑀 ) ) )
6 2 5 rspcedv ( ( 𝑅𝑈𝑁𝑉𝑀𝑁 ) → ( 𝑥 ∈ ( 𝑅 𝑀 ) → ∃ 𝑛𝑁 𝑥 ∈ ( 𝑅 𝑛 ) ) )
7 1 eliunov2 ( ( 𝑅𝑈𝑁𝑉 ) → ( 𝑥 ∈ ( 𝐶𝑅 ) ↔ ∃ 𝑛𝑁 𝑥 ∈ ( 𝑅 𝑛 ) ) )
8 7 biimprd ( ( 𝑅𝑈𝑁𝑉 ) → ( ∃ 𝑛𝑁 𝑥 ∈ ( 𝑅 𝑛 ) → 𝑥 ∈ ( 𝐶𝑅 ) ) )
9 8 3adant3 ( ( 𝑅𝑈𝑁𝑉𝑀𝑁 ) → ( ∃ 𝑛𝑁 𝑥 ∈ ( 𝑅 𝑛 ) → 𝑥 ∈ ( 𝐶𝑅 ) ) )
10 6 9 syld ( ( 𝑅𝑈𝑁𝑉𝑀𝑁 ) → ( 𝑥 ∈ ( 𝑅 𝑀 ) → 𝑥 ∈ ( 𝐶𝑅 ) ) )
11 10 ssrdv ( ( 𝑅𝑈𝑁𝑉𝑀𝑁 ) → ( 𝑅 𝑀 ) ⊆ ( 𝐶𝑅 ) )