Theorem reldmtng 24969

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 24915	. 2 ⊢ toNrmGrp = (𝑔 ∈ V, 𝑓 ∈ V ↦ ((𝑔 sSet ⟨(dist‘ndx), (𝑓 ∘ (-g‘𝑔))⟩) sSet ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-g‘𝑔)))⟩))
2	1	reldmmpo 7561	1 ⊢ Rel dom toNrmGrp

This proof depends on syntax axioms:  Vcvv 3453  ⟨cop 4837  dom cdm 5654   ∘ ccom 5658  Rel wrel 5659  ‘cfv 6546  (class class class)co 7427   sSet csts 17353  ndxcnx 17383  TopSetcts 17446  distcds 17449  -_gcsg 19158  MetOpencmopn 21680   toNrmGrp ctng 24909

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-br 5106  df-opab 5170  df-xp 5660  df-rel 5661  df-dm 5664  df-oprab 7431  df-mpo 7432  df-tng 24915

This theorem is used by:  tnglem  24971  tngds  24979  tcphval  25551