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Theorem pstmval 32345
Description: Value of the metric induced by a pseudometric 𝐷. (Contributed by Thierry Arnoux, 7-Feb-2018.)
Hypothesis
Ref Expression
pstmval.1 = (~Met𝐷)
Assertion
Ref Expression
pstmval (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) = (𝑎 ∈ (𝑋 / ), 𝑏 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝐷𝑦)}))
Distinct variable groups:   𝑎,𝑏,𝑥,𝑦,𝑧,𝐷   𝑋,𝑎,𝑏,𝑥,𝑦,𝑧   ,𝑎,𝑏,𝑥,𝑦,𝑧

Proof of Theorem pstmval
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 df-pstm 32339 . 2 pstoMet = (𝑑 ran PsMet ↦ (𝑎 ∈ (dom dom 𝑑 / (~Met𝑑)), 𝑏 ∈ (dom dom 𝑑 / (~Met𝑑)) ↦ {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝑑𝑦)}))
2 psmetdmdm 23642 . . . . . . . 8 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 = dom dom 𝐷)
32adantr 481 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → 𝑋 = dom dom 𝐷)
4 dmeq 5857 . . . . . . . . 9 (𝑑 = 𝐷 → dom 𝑑 = dom 𝐷)
54dmeqd 5859 . . . . . . . 8 (𝑑 = 𝐷 → dom dom 𝑑 = dom dom 𝐷)
65adantl 482 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = dom dom 𝐷)
73, 6eqtr4d 2779 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → 𝑋 = dom dom 𝑑)
8 qseq1 8698 . . . . . 6 (𝑋 = dom dom 𝑑 → (𝑋 / ) = (dom dom 𝑑 / ))
97, 8syl 17 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑋 / ) = (dom dom 𝑑 / ))
10 pstmval.1 . . . . . . . 8 = (~Met𝐷)
11 fveq2 6839 . . . . . . . 8 (𝑑 = 𝐷 → (~Met𝑑) = (~Met𝐷))
1210, 11eqtr4id 2795 . . . . . . 7 (𝑑 = 𝐷 = (~Met𝑑))
1312qseq2d 8701 . . . . . 6 (𝑑 = 𝐷 → (dom dom 𝑑 / ) = (dom dom 𝑑 / (~Met𝑑)))
1413adantl 482 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (dom dom 𝑑 / ) = (dom dom 𝑑 / (~Met𝑑)))
159, 14eqtr2d 2777 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (dom dom 𝑑 / (~Met𝑑)) = (𝑋 / ))
16 mpoeq12 7426 . . . 4 (((dom dom 𝑑 / (~Met𝑑)) = (𝑋 / ) ∧ (dom dom 𝑑 / (~Met𝑑)) = (𝑋 / )) → (𝑎 ∈ (dom dom 𝑑 / (~Met𝑑)), 𝑏 ∈ (dom dom 𝑑 / (~Met𝑑)) ↦ {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝑑𝑦)}) = (𝑎 ∈ (𝑋 / ), 𝑏 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝑑𝑦)}))
1715, 15, 16syl2anc 584 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑎 ∈ (dom dom 𝑑 / (~Met𝑑)), 𝑏 ∈ (dom dom 𝑑 / (~Met𝑑)) ↦ {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝑑𝑦)}) = (𝑎 ∈ (𝑋 / ), 𝑏 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝑑𝑦)}))
18 simp1r 1198 . . . . . . . . 9 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ (𝑋 / ) ∧ 𝑏 ∈ (𝑋 / )) → 𝑑 = 𝐷)
1918oveqd 7370 . . . . . . . 8 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ (𝑋 / ) ∧ 𝑏 ∈ (𝑋 / )) → (𝑥𝑑𝑦) = (𝑥𝐷𝑦))
2019eqeq2d 2747 . . . . . . 7 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ (𝑋 / ) ∧ 𝑏 ∈ (𝑋 / )) → (𝑧 = (𝑥𝑑𝑦) ↔ 𝑧 = (𝑥𝐷𝑦)))
21202rexbidv 3211 . . . . . 6 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ (𝑋 / ) ∧ 𝑏 ∈ (𝑋 / )) → (∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝑑𝑦) ↔ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝐷𝑦)))
2221abbidv 2805 . . . . 5 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ (𝑋 / ) ∧ 𝑏 ∈ (𝑋 / )) → {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝑑𝑦)} = {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝐷𝑦)})
2322unieqd 4877 . . . 4 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ (𝑋 / ) ∧ 𝑏 ∈ (𝑋 / )) → {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝑑𝑦)} = {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝐷𝑦)})
2423mpoeq3dva 7430 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑎 ∈ (𝑋 / ), 𝑏 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝑑𝑦)}) = (𝑎 ∈ (𝑋 / ), 𝑏 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝐷𝑦)}))
2517, 24eqtrd 2776 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑎 ∈ (dom dom 𝑑 / (~Met𝑑)), 𝑏 ∈ (dom dom 𝑑 / (~Met𝑑)) ↦ {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝑑𝑦)}) = (𝑎 ∈ (𝑋 / ), 𝑏 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝐷𝑦)}))
26 elfvunirn 6871 . 2 (𝐷 ∈ (PsMet‘𝑋) → 𝐷 ran PsMet)
27 elfvex 6877 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
28 qsexg 8710 . . . 4 (𝑋 ∈ V → (𝑋 / ) ∈ V)
2927, 28syl 17 . . 3 (𝐷 ∈ (PsMet‘𝑋) → (𝑋 / ) ∈ V)
30 mpoexga 8006 . . 3 (((𝑋 / ) ∈ V ∧ (𝑋 / ) ∈ V) → (𝑎 ∈ (𝑋 / ), 𝑏 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝐷𝑦)}) ∈ V)
3129, 29, 30syl2anc 584 . 2 (𝐷 ∈ (PsMet‘𝑋) → (𝑎 ∈ (𝑋 / ), 𝑏 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝐷𝑦)}) ∈ V)
321, 25, 26, 31fvmptd2 6953 1 (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) = (𝑎 ∈ (𝑋 / ), 𝑏 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝐷𝑦)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  {cab 2713  wrex 3071  Vcvv 3443   cuni 4863  dom cdm 5631  ran crn 5632  cfv 6493  (class class class)co 7353  cmpo 7355   / cqs 8643  PsMetcpsmet 20765  ~Metcmetid 32336  pstoMetcpstm 32337
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7668  ax-cnex 11103  ax-resscn 11104
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7917  df-2nd 7918  df-ec 8646  df-qs 8650  df-map 8763  df-xr 11189  df-psmet 20773  df-pstm 32339
This theorem is referenced by:  pstmfval  32346  pstmxmet  32347
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