Metamath Proof Explorer

Theorem elcnfn

Description: Property defining a continuous functional. (Contributed by NM, 11-Feb-2006) (Revised by Mario Carneiro, 16-Nov-2013) (New usage is discouraged.)

Ref Expression
Assertion elcnfn ⊢T∈ContFn↔T:ℋ⟶ℂ∧∀x∈ℋ∀y∈ℝ+∃z∈ℝ+∀w∈ℋnormℎ⁡w-ℎx<z→T⁡w−T⁡x<y

Proof

Step Hyp Ref Expression
1 fveq1 ⊢t=T→t⁡w=T⁡w
2 fveq1 ⊢t=T→t⁡x=T⁡x
3 1 2 oveq12d ⊢t=T→t⁡w−t⁡x=T⁡w−T⁡x
4 3 fveq2d ⊢t=T→t⁡w−t⁡x=T⁡w−T⁡x
5 4 breq1d ⊢t=T→t⁡w−t⁡x<y↔T⁡w−T⁡x<y
6 5 imbi2d ⊢t=T→normℎ⁡w-ℎx<z→t⁡w−t⁡x<y↔normℎ⁡w-ℎx<z→T⁡w−T⁡x<y
7 6 rexralbidv ⊢t=T→∃z∈ℝ+∀w∈ℋnormℎ⁡w-ℎx<z→t⁡w−t⁡x<y↔∃z∈ℝ+∀w∈ℋnormℎ⁡w-ℎx<z→T⁡w−T⁡x<y
8 7 2ralbidv ⊢t=T→∀x∈ℋ∀y∈ℝ+∃z∈ℝ+∀w∈ℋnormℎ⁡w-ℎx<z→t⁡w−t⁡x<y↔∀x∈ℋ∀y∈ℝ+∃z∈ℝ+∀w∈ℋnormℎ⁡w-ℎx<z→T⁡w−T⁡x<y
9 df-cnfn ⊢ContFn=t∈ℂℋ|∀x∈ℋ∀y∈ℝ+∃z∈ℝ+∀w∈ℋnormℎ⁡w-ℎx<z→t⁡w−t⁡x<y
10 8 9 elrab2 ⊢T∈ContFn↔T∈ℂℋ∧∀x∈ℋ∀y∈ℝ+∃z∈ℝ+∀w∈ℋnormℎ⁡w-ℎx<z→T⁡w−T⁡x<y
11 cnex ⊢ℂ∈V
12 ax-hilex ⊢ℋ∈V
13 11 12 elmap ⊢T∈ℂℋ↔T:ℋ⟶ℂ
14 13 anbi1i ⊢T∈ℂℋ∧∀x∈ℋ∀y∈ℝ+∃z∈ℝ+∀w∈ℋnormℎ⁡w-ℎx<z→T⁡w−T⁡x<y↔T:ℋ⟶ℂ∧∀x∈ℋ∀y∈ℝ+∃z∈ℝ+∀w∈ℋnormℎ⁡w-ℎx<z→T⁡w−T⁡x<y
15 10 14 bitri ⊢T∈ContFn↔T:ℋ⟶ℂ∧∀x∈ℋ∀y∈ℝ+∃z∈ℝ+∀w∈ℋnormℎ⁡w-ℎx<z→T⁡w−T⁡x<y