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Theorem ismet 22350
Description: Express the predicate "𝐷 is a metric." (Contributed by NM, 25-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
Assertion
Ref Expression
ismet (𝑋𝐴 → (𝐷 ∈ (Met‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ ∧ ∀𝑥𝑋𝑦𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦))))))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐷   𝑥,𝑋,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧)

Proof of Theorem ismet
Dummy variables 𝑑 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3353 . . . . 5 (𝑋𝐴𝑋 ∈ V)
2 xpeq12 5292 . . . . . . . . 9 ((𝑡 = 𝑋𝑡 = 𝑋) → (𝑡 × 𝑡) = (𝑋 × 𝑋))
32anidms 680 . . . . . . . 8 (𝑡 = 𝑋 → (𝑡 × 𝑡) = (𝑋 × 𝑋))
43oveq2d 6831 . . . . . . 7 (𝑡 = 𝑋 → (ℝ ↑𝑚 (𝑡 × 𝑡)) = (ℝ ↑𝑚 (𝑋 × 𝑋)))
5 raleq 3278 . . . . . . . . . 10 (𝑡 = 𝑋 → (∀𝑧𝑡 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)) ↔ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦))))
65anbi2d 742 . . . . . . . . 9 (𝑡 = 𝑋 → ((((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑡 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦))) ↔ (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))))
76raleqbi1dv 3286 . . . . . . . 8 (𝑡 = 𝑋 → (∀𝑦𝑡 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑡 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦))) ↔ ∀𝑦𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))))
87raleqbi1dv 3286 . . . . . . 7 (𝑡 = 𝑋 → (∀𝑥𝑡𝑦𝑡 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑡 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦))) ↔ ∀𝑥𝑋𝑦𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))))
94, 8rabeqbidv 3336 . . . . . 6 (𝑡 = 𝑋 → {𝑑 ∈ (ℝ ↑𝑚 (𝑡 × 𝑡)) ∣ ∀𝑥𝑡𝑦𝑡 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑡 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))} = {𝑑 ∈ (ℝ ↑𝑚 (𝑋 × 𝑋)) ∣ ∀𝑥𝑋𝑦𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))})
10 df-met 19963 . . . . . 6 Met = (𝑡 ∈ V ↦ {𝑑 ∈ (ℝ ↑𝑚 (𝑡 × 𝑡)) ∣ ∀𝑥𝑡𝑦𝑡 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑡 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))})
11 ovex 6843 . . . . . . 7 (ℝ ↑𝑚 (𝑋 × 𝑋)) ∈ V
1211rabex 4965 . . . . . 6 {𝑑 ∈ (ℝ ↑𝑚 (𝑋 × 𝑋)) ∣ ∀𝑥𝑋𝑦𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))} ∈ V
139, 10, 12fvmpt 6446 . . . . 5 (𝑋 ∈ V → (Met‘𝑋) = {𝑑 ∈ (ℝ ↑𝑚 (𝑋 × 𝑋)) ∣ ∀𝑥𝑋𝑦𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))})
141, 13syl 17 . . . 4 (𝑋𝐴 → (Met‘𝑋) = {𝑑 ∈ (ℝ ↑𝑚 (𝑋 × 𝑋)) ∣ ∀𝑥𝑋𝑦𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))})
1514eleq2d 2826 . . 3 (𝑋𝐴 → (𝐷 ∈ (Met‘𝑋) ↔ 𝐷 ∈ {𝑑 ∈ (ℝ ↑𝑚 (𝑋 × 𝑋)) ∣ ∀𝑥𝑋𝑦𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))}))
16 oveq 6821 . . . . . . . 8 (𝑑 = 𝐷 → (𝑥𝑑𝑦) = (𝑥𝐷𝑦))
1716eqeq1d 2763 . . . . . . 7 (𝑑 = 𝐷 → ((𝑥𝑑𝑦) = 0 ↔ (𝑥𝐷𝑦) = 0))
1817bibi1d 332 . . . . . 6 (𝑑 = 𝐷 → (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ↔ ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦)))
19 oveq 6821 . . . . . . . . 9 (𝑑 = 𝐷 → (𝑧𝑑𝑥) = (𝑧𝐷𝑥))
20 oveq 6821 . . . . . . . . 9 (𝑑 = 𝐷 → (𝑧𝑑𝑦) = (𝑧𝐷𝑦))
2119, 20oveq12d 6833 . . . . . . . 8 (𝑑 = 𝐷 → ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)) = ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))
2216, 21breq12d 4818 . . . . . . 7 (𝑑 = 𝐷 → ((𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)) ↔ (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦))))
2322ralbidv 3125 . . . . . 6 (𝑑 = 𝐷 → (∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)) ↔ ∀𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦))))
2418, 23anbi12d 749 . . . . 5 (𝑑 = 𝐷 → ((((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦))) ↔ (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))))
25242ralbidv 3128 . . . 4 (𝑑 = 𝐷 → (∀𝑥𝑋𝑦𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦))) ↔ ∀𝑥𝑋𝑦𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))))
2625elrab 3505 . . 3 (𝐷 ∈ {𝑑 ∈ (ℝ ↑𝑚 (𝑋 × 𝑋)) ∣ ∀𝑥𝑋𝑦𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))} ↔ (𝐷 ∈ (ℝ ↑𝑚 (𝑋 × 𝑋)) ∧ ∀𝑥𝑋𝑦𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))))
2715, 26syl6bb 276 . 2 (𝑋𝐴 → (𝐷 ∈ (Met‘𝑋) ↔ (𝐷 ∈ (ℝ ↑𝑚 (𝑋 × 𝑋)) ∧ ∀𝑥𝑋𝑦𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦))))))
28 reex 10240 . . . 4 ℝ ∈ V
29 sqxpexg 7130 . . . 4 (𝑋𝐴 → (𝑋 × 𝑋) ∈ V)
30 elmapg 8039 . . . 4 ((ℝ ∈ V ∧ (𝑋 × 𝑋) ∈ V) → (𝐷 ∈ (ℝ ↑𝑚 (𝑋 × 𝑋)) ↔ 𝐷:(𝑋 × 𝑋)⟶ℝ))
3128, 29, 30sylancr 698 . . 3 (𝑋𝐴 → (𝐷 ∈ (ℝ ↑𝑚 (𝑋 × 𝑋)) ↔ 𝐷:(𝑋 × 𝑋)⟶ℝ))
3231anbi1d 743 . 2 (𝑋𝐴 → ((𝐷 ∈ (ℝ ↑𝑚 (𝑋 × 𝑋)) ∧ ∀𝑥𝑋𝑦𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ ∧ ∀𝑥𝑋𝑦𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦))))))
3327, 32bitrd 268 1 (𝑋𝐴 → (𝐷 ∈ (Met‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ ∧ ∀𝑥𝑋𝑦𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2140  wral 3051  {crab 3055  Vcvv 3341   class class class wbr 4805   × cxp 5265  wf 6046  cfv 6050  (class class class)co 6815  𝑚 cmap 8026  cr 10148  0cc0 10149   + caddc 10152  cle 10288  Metcme 19955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116  ax-cnex 10205  ax-resscn 10206
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-fv 6058  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-map 8028  df-met 19963
This theorem is referenced by:  ismeti  22352  metflem  22355  ismet2  22360  dscmet  22599  nrmmetd  22601  rrxmet  23412  metf1o  33883  rrnmet  33960
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