MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reldmtng Structured version   Visualization version   GIF version

Theorem reldmtng 23242
Description: The function toNrmGrp is a two-argument function. (Contributed by Mario Carneiro, 8-Oct-2015.)
Assertion
Ref Expression
reldmtng Rel dom toNrmGrp

Proof of Theorem reldmtng
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-tng 23189 . 2 toNrmGrp = (𝑔 ∈ V, 𝑓 ∈ V ↦ ((𝑔 sSet ⟨(dist‘ndx), (𝑓 ∘ (-g𝑔))⟩) sSet ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-g𝑔)))⟩))
21reldmmpo 7269 1 Rel dom toNrmGrp
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3469  cop 4545  dom cdm 5532  ccom 5536  Rel wrel 5537  cfv 6334  (class class class)co 7140  ndxcnx 16471   sSet csts 16472  TopSetcts 16562  distcds 16565  -gcsg 18096  MetOpencmopn 20079   toNrmGrp ctng 23183
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-xp 5538  df-rel 5539  df-dm 5542  df-oprab 7144  df-mpo 7145  df-tng 23189
This theorem is referenced by:  tnglem  23244  tngds  23252  tcphval  23820
  Copyright terms: Public domain W3C validator