Theorem fntopon 22073

Description: The class TopOn is a function with domain the universal class V. Analogue for topologies of fnmre 17300 for Moore collections. (Contributed by BJ, 29-Apr-2021.)

Assertion

Ref	Expression
fntopon	⊢ TopOn Fn V

Proof of Theorem fntopon

Step	Hyp	Ref	Expression
1		funtopon 22069	. 2 ⊢ Fun TopOn
2		dmtopon 22072	. 2 ⊢ dom TopOn = V
3		df-fn 6436	. 2 ⊢ (TopOn Fn V ↔ (Fun TopOn ∧ dom TopOn = V))
4	1, 2, 3	mpbir2an 708	1 ⊢ TopOn Fn V

Syntax hints:   = wceq 1539  Vcvv 3432  dom cdm 5589  Fun wfun 6427   Fn wfn 6428  TopOnctopon 22059

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-pow 5288  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-ral 3069  df-rex 3070  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-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-fun 6435  df-fn 6436  df-topon 22060

This theorem is referenced by:  toprntopon  22074