Metamath Proof Explorer

Theorem isms

Description: Express the predicate " <. X , D >. is a metric space" with underlying set X and distance function D . (Contributed by NM, 27-Aug-2006) (Revised by Mario Carneiro, 24-Aug-2015)

Ref Expression
Hypotheses isms.j ⊒ 𝐽 = ( TopOpen β€˜ 𝐾 )
isms.x ⊒ 𝑋 = ( Base β€˜ 𝐾 )
isms.d ⊒ 𝐷 = ( ( dist β€˜ 𝐾 ) β†Ύ ( 𝑋 Γ— 𝑋 ) )
Assertion isms ( 𝐾 ∈ MetSp ↔ ( 𝐾 ∈ ∞MetSp ∧ 𝐷 ∈ ( Met β€˜ 𝑋 ) ) )

Proof

Step Hyp Ref Expression
1 isms.j ⊒ 𝐽 = ( TopOpen β€˜ 𝐾 )
2 isms.x ⊒ 𝑋 = ( Base β€˜ 𝐾 )
3 isms.d ⊒ 𝐷 = ( ( dist β€˜ 𝐾 ) β†Ύ ( 𝑋 Γ— 𝑋 ) )
4 fveq2 ⊒ ( 𝑓 = 𝐾 β†’ ( dist β€˜ 𝑓 ) = ( dist β€˜ 𝐾 ) )
5 fveq2 ⊒ ( 𝑓 = 𝐾 β†’ ( Base β€˜ 𝑓 ) = ( Base β€˜ 𝐾 ) )
6 5 2 eqtr4di ⊒ ( 𝑓 = 𝐾 β†’ ( Base β€˜ 𝑓 ) = 𝑋 )
7 6 sqxpeqd ⊒ ( 𝑓 = 𝐾 β†’ ( ( Base β€˜ 𝑓 ) Γ— ( Base β€˜ 𝑓 ) ) = ( 𝑋 Γ— 𝑋 ) )
8 4 7 reseq12d ⊒ ( 𝑓 = 𝐾 β†’ ( ( dist β€˜ 𝑓 ) β†Ύ ( ( Base β€˜ 𝑓 ) Γ— ( Base β€˜ 𝑓 ) ) ) = ( ( dist β€˜ 𝐾 ) β†Ύ ( 𝑋 Γ— 𝑋 ) ) )
9 8 3 eqtr4di ⊒ ( 𝑓 = 𝐾 β†’ ( ( dist β€˜ 𝑓 ) β†Ύ ( ( Base β€˜ 𝑓 ) Γ— ( Base β€˜ 𝑓 ) ) ) = 𝐷 )
10 6 fveq2d ⊒ ( 𝑓 = 𝐾 β†’ ( Met β€˜ ( Base β€˜ 𝑓 ) ) = ( Met β€˜ 𝑋 ) )
11 9 10 eleq12d ⊒ ( 𝑓 = 𝐾 β†’ ( ( ( dist β€˜ 𝑓 ) β†Ύ ( ( Base β€˜ 𝑓 ) Γ— ( Base β€˜ 𝑓 ) ) ) ∈ ( Met β€˜ ( Base β€˜ 𝑓 ) ) ↔ 𝐷 ∈ ( Met β€˜ 𝑋 ) ) )
12 df-ms ⊒ MetSp = { 𝑓 ∈ ∞MetSp ∣ ( ( dist β€˜ 𝑓 ) β†Ύ ( ( Base β€˜ 𝑓 ) Γ— ( Base β€˜ 𝑓 ) ) ) ∈ ( Met β€˜ ( Base β€˜ 𝑓 ) ) }
13 11 12 elrab2 ⊒ ( 𝐾 ∈ MetSp ↔ ( 𝐾 ∈ ∞MetSp ∧ 𝐷 ∈ ( Met β€˜ 𝑋 ) ) )