Theorem reldmtng 23794

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 23740	. 2 ⊢ toNrmGrp = (𝑔 ∈ V, 𝑓 ∈ V ↦ ((𝑔 sSet ⟨(dist‘ndx), (𝑓 ∘ (-g‘𝑔))⟩) sSet ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-g‘𝑔)))⟩))
2	1	reldmmpo 7408	1 ⊢ Rel dom toNrmGrp

Syntax hints:  Vcvv 3432  ⟨cop 4567  dom cdm 5589   ∘ ccom 5593  Rel wrel 5594  ‘cfv 6433  (class class class)co 7275   sSet csts 16864  ndxcnx 16894  TopSetcts 16968  distcds 16971  -_gcsg 18579  MetOpencmopn 20587   toNrmGrp ctng 23734

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-dm 5599  df-oprab 7279  df-mpo 7280  df-tng 23740

This theorem is referenced by:  tnglem  23796  tnglemOLD  23797  tngds  23811  tngdsOLD  23812  tcphval  24382