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Metamath Proof Explorer

Theorem isucn

Description: The predicate " F is a uniformly continuous function from uniform space U to uniform space V ". (Contributed by Thierry Arnoux, 16-Nov-2017)

Ref Expression
Assertion isucn ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑉 ∈ ( UnifOn ‘ 𝑌 ) ) → ( 𝐹 ∈ ( 𝑈 Cnu 𝑉 ) ↔ ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑠 ∈ 𝑉 ∃ 𝑟 ∈ 𝑈 ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝑥 𝑟 𝑦 → ( 𝐹 ‘ 𝑥 ) 𝑠 ( 𝐹 ‘ 𝑦 ) ) ) ) )

Proof

Step Hyp Ref Expression
1 ucnval ⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑉 ∈ ( UnifOn ‘ 𝑌 ) ) → ( 𝑈 Cnu 𝑉 ) = { 𝑓 ∈ ( 𝑌 ↑m 𝑋 ) ∣ ∀ 𝑠 ∈ 𝑉 ∃ 𝑟 ∈ 𝑈 ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝑥 𝑟 𝑦 → ( 𝑓 ‘ 𝑥 ) 𝑠 ( 𝑓 ‘ 𝑦 ) ) } )
2 1 eleq2d ⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑉 ∈ ( UnifOn ‘ 𝑌 ) ) → ( 𝐹 ∈ ( 𝑈 Cnu 𝑉 ) ↔ 𝐹 ∈ { 𝑓 ∈ ( 𝑌 ↑m 𝑋 ) ∣ ∀ 𝑠 ∈ 𝑉 ∃ 𝑟 ∈ 𝑈 ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝑥 𝑟 𝑦 → ( 𝑓 ‘ 𝑥 ) 𝑠 ( 𝑓 ‘ 𝑦 ) ) } ) )
3 fveq1 ⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) )
4 fveq1 ⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) )
5 3 4 breq12d ⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 ‘ 𝑥 ) 𝑠 ( 𝑓 ‘ 𝑦 ) ↔ ( 𝐹 ‘ 𝑥 ) 𝑠 ( 𝐹 ‘ 𝑦 ) ) )
6 5 imbi2d ⊢ ( 𝑓 = 𝐹 → ( ( 𝑥 𝑟 𝑦 → ( 𝑓 ‘ 𝑥 ) 𝑠 ( 𝑓 ‘ 𝑦 ) ) ↔ ( 𝑥 𝑟 𝑦 → ( 𝐹 ‘ 𝑥 ) 𝑠 ( 𝐹 ‘ 𝑦 ) ) ) )
7 6 ralbidv ⊢ ( 𝑓 = 𝐹 → ( ∀ 𝑦 ∈ 𝑋 ( 𝑥 𝑟 𝑦 → ( 𝑓 ‘ 𝑥 ) 𝑠 ( 𝑓 ‘ 𝑦 ) ) ↔ ∀ 𝑦 ∈ 𝑋 ( 𝑥 𝑟 𝑦 → ( 𝐹 ‘ 𝑥 ) 𝑠 ( 𝐹 ‘ 𝑦 ) ) ) )
8 7 rexralbidv ⊢ ( 𝑓 = 𝐹 → ( ∃ 𝑟 ∈ 𝑈 ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝑥 𝑟 𝑦 → ( 𝑓 ‘ 𝑥 ) 𝑠 ( 𝑓 ‘ 𝑦 ) ) ↔ ∃ 𝑟 ∈ 𝑈 ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝑥 𝑟 𝑦 → ( 𝐹 ‘ 𝑥 ) 𝑠 ( 𝐹 ‘ 𝑦 ) ) ) )
9 8 ralbidv ⊢ ( 𝑓 = 𝐹 → ( ∀ 𝑠 ∈ 𝑉 ∃ 𝑟 ∈ 𝑈 ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝑥 𝑟 𝑦 → ( 𝑓 ‘ 𝑥 ) 𝑠 ( 𝑓 ‘ 𝑦 ) ) ↔ ∀ 𝑠 ∈ 𝑉 ∃ 𝑟 ∈ 𝑈 ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝑥 𝑟 𝑦 → ( 𝐹 ‘ 𝑥 ) 𝑠 ( 𝐹 ‘ 𝑦 ) ) ) )
10 9 elrab ⊢ ( 𝐹 ∈ { 𝑓 ∈ ( 𝑌 ↑m 𝑋 ) ∣ ∀ 𝑠 ∈ 𝑉 ∃ 𝑟 ∈ 𝑈 ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝑥 𝑟 𝑦 → ( 𝑓 ‘ 𝑥 ) 𝑠 ( 𝑓 ‘ 𝑦 ) ) } ↔ ( 𝐹 ∈ ( 𝑌 ↑m 𝑋 ) ∧ ∀ 𝑠 ∈ 𝑉 ∃ 𝑟 ∈ 𝑈 ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝑥 𝑟 𝑦 → ( 𝐹 ‘ 𝑥 ) 𝑠 ( 𝐹 ‘ 𝑦 ) ) ) )
11 2 10 bitrdi ⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑉 ∈ ( UnifOn ‘ 𝑌 ) ) → ( 𝐹 ∈ ( 𝑈 Cnu 𝑉 ) ↔ ( 𝐹 ∈ ( 𝑌 ↑m 𝑋 ) ∧ ∀ 𝑠 ∈ 𝑉 ∃ 𝑟 ∈ 𝑈 ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝑥 𝑟 𝑦 → ( 𝐹 ‘ 𝑥 ) 𝑠 ( 𝐹 ‘ 𝑦 ) ) ) ) )
12 elfvex ⊢ ( 𝑉 ∈ ( UnifOn ‘ 𝑌 ) → 𝑌 ∈ V )
13 elfvex ⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → 𝑋 ∈ V )
14 elmapg ⊢ ( ( 𝑌 ∈ V ∧ 𝑋 ∈ V ) → ( 𝐹 ∈ ( 𝑌 ↑m 𝑋 ) ↔ 𝐹 : 𝑋 ⟶ 𝑌 ) )
15 12 13 14 syl2anr ⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑉 ∈ ( UnifOn ‘ 𝑌 ) ) → ( 𝐹 ∈ ( 𝑌 ↑m 𝑋 ) ↔ 𝐹 : 𝑋 ⟶ 𝑌 ) )
16 15 anbi1d ⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑉 ∈ ( UnifOn ‘ 𝑌 ) ) → ( ( 𝐹 ∈ ( 𝑌 ↑m 𝑋 ) ∧ ∀ 𝑠 ∈ 𝑉 ∃ 𝑟 ∈ 𝑈 ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝑥 𝑟 𝑦 → ( 𝐹 ‘ 𝑥 ) 𝑠 ( 𝐹 ‘ 𝑦 ) ) ) ↔ ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑠 ∈ 𝑉 ∃ 𝑟 ∈ 𝑈 ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝑥 𝑟 𝑦 → ( 𝐹 ‘ 𝑥 ) 𝑠 ( 𝐹 ‘ 𝑦 ) ) ) ) )
17 11 16 bitrd ⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑉 ∈ ( UnifOn ‘ 𝑌 ) ) → ( 𝐹 ∈ ( 𝑈 Cnu 𝑉 ) ↔ ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑠 ∈ 𝑉 ∃ 𝑟 ∈ 𝑈 ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝑥 𝑟 𝑦 → ( 𝐹 ‘ 𝑥 ) 𝑠 ( 𝐹 ‘ 𝑦 ) ) ) ) )