Theorem reldmtng 24116

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 24062	. 2 ⊢ toNrmGrp = (𝑔 ∈ V, 𝑓 ∈ V ↦ ((𝑔 sSet ⟨(dist‘ndx), (𝑓 ∘ (-g‘𝑔))⟩) sSet ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-g‘𝑔)))⟩))
2	1	reldmmpo 7530	1 ⊢ Rel dom toNrmGrp

Syntax hints:  Vcvv 3475  ⟨cop 4630  dom cdm 5672   ∘ ccom 5676  Rel wrel 5677  ‘cfv 6535  (class class class)co 7396   sSet csts 17083  ndxcnx 17113  TopSetcts 17190  distcds 17193  -_gcsg 18808  MetOpencmopn 20908   toNrmGrp ctng 24056

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5145  df-opab 5207  df-xp 5678  df-rel 5679  df-dm 5682  df-oprab 7400  df-mpo 7401  df-tng 24062

This theorem is referenced by:  tnglem  24118  tnglemOLD  24119  tngds  24133  tngdsOLD  24134  tcphval  24704