Theorem reldmtng 23700

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.

Step	Hyp	Ref	Expression
1		df-tng 23646	. 2 ⊢ toNrmGrp = (𝑔 ∈ V, 𝑓 ∈ V ↦ ((𝑔 sSet ⟨(dist‘ndx), (𝑓 ∘ (-g‘𝑔))⟩) sSet ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-g‘𝑔)))⟩))
2	1	reldmmpo 7386	1 ⊢ Rel dom toNrmGrp

Syntax hints:  Vcvv 3422  ⟨cop 4564  dom cdm 5580   ∘ ccom 5584  Rel wrel 5585  ‘cfv 6418  (class class class)co 7255   sSet csts 16792  ndxcnx 16822  TopSetcts 16894  distcds 16897  -_gcsg 18494  MetOpencmopn 20500   toNrmGrp ctng 23640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-dm 5590  df-oprab 7259  df-mpo 7260  df-tng 23646

This theorem is referenced by:  tnglem  23702  tnglemOLD  23703  tngds  23717  tngdsOLD  23718  tcphval  24287