Metamath Proof Explorer

Theorem isxms

Description: Express the predicate " <. X , D >. is an extended metric space" with underlying set X and distance function D . (Contributed by Mario Carneiro, 2-Sep-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 isxms ( 𝐾 ∈ ∞MetSp ↔ ( 𝐾 ∈ TopSp ∧ 𝐽 = ( MetOpen β€˜ 𝐷 ) ) )

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†’ ( TopOpen β€˜ 𝑓 ) = ( TopOpen β€˜ 𝐾 ) )
5 4 1 eqtr4di ⊒ ( 𝑓 = 𝐾 β†’ ( TopOpen β€˜ 𝑓 ) = 𝐽 )
6 fveq2 ⊒ ( 𝑓 = 𝐾 β†’ ( dist β€˜ 𝑓 ) = ( dist β€˜ 𝐾 ) )
7 fveq2 ⊒ ( 𝑓 = 𝐾 β†’ ( Base β€˜ 𝑓 ) = ( Base β€˜ 𝐾 ) )
8 7 2 eqtr4di ⊒ ( 𝑓 = 𝐾 β†’ ( Base β€˜ 𝑓 ) = 𝑋 )
9 8 sqxpeqd ⊒ ( 𝑓 = 𝐾 β†’ ( ( Base β€˜ 𝑓 ) Γ— ( Base β€˜ 𝑓 ) ) = ( 𝑋 Γ— 𝑋 ) )
10 6 9 reseq12d ⊒ ( 𝑓 = 𝐾 β†’ ( ( dist β€˜ 𝑓 ) β†Ύ ( ( Base β€˜ 𝑓 ) Γ— ( Base β€˜ 𝑓 ) ) ) = ( ( dist β€˜ 𝐾 ) β†Ύ ( 𝑋 Γ— 𝑋 ) ) )
11 10 3 eqtr4di ⊒ ( 𝑓 = 𝐾 β†’ ( ( dist β€˜ 𝑓 ) β†Ύ ( ( Base β€˜ 𝑓 ) Γ— ( Base β€˜ 𝑓 ) ) ) = 𝐷 )
12 11 fveq2d ⊒ ( 𝑓 = 𝐾 β†’ ( MetOpen β€˜ ( ( dist β€˜ 𝑓 ) β†Ύ ( ( Base β€˜ 𝑓 ) Γ— ( Base β€˜ 𝑓 ) ) ) ) = ( MetOpen β€˜ 𝐷 ) )
13 5 12 eqeq12d ⊒ ( 𝑓 = 𝐾 β†’ ( ( TopOpen β€˜ 𝑓 ) = ( MetOpen β€˜ ( ( dist β€˜ 𝑓 ) β†Ύ ( ( Base β€˜ 𝑓 ) Γ— ( Base β€˜ 𝑓 ) ) ) ) ↔ 𝐽 = ( MetOpen β€˜ 𝐷 ) ) )
14 df-xms ⊒ ∞MetSp = { 𝑓 ∈ TopSp ∣ ( TopOpen β€˜ 𝑓 ) = ( MetOpen β€˜ ( ( dist β€˜ 𝑓 ) β†Ύ ( ( Base β€˜ 𝑓 ) Γ— ( Base β€˜ 𝑓 ) ) ) ) }
15 13 14 elrab2 ⊒ ( 𝐾 ∈ ∞MetSp ↔ ( 𝐾 ∈ TopSp ∧ 𝐽 = ( MetOpen β€˜ 𝐷 ) ) )