Theorem topontopi 22784

Description: A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)

Hypothesis

Ref	Expression
topontopi.1	⊢ 𝐽 ∈ (TopOn‘𝐵)

Assertion

Ref	Expression
topontopi	⊢ 𝐽 ∈ Top

Proof of Theorem topontopi

Step	Hyp	Ref	Expression
1		topontopi.1	. 2 ⊢ 𝐽 ∈ (TopOn‘𝐵)
2		topontop 22782	. 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
3	1, 2	ax-mp 5	1 ⊢ 𝐽 ∈ Top

Syntax hints:   ∈ wcel 2109  ‘cfv 6476  Topctop 22762  TopOnctopon 22779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5367  ax-un 7662

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3393  df-v 3435  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5089  df-opab 5151  df-mpt 5170  df-id 5508  df-xp 5619  df-rel 5620  df-cnv 5621  df-co 5622  df-dm 5623  df-iota 6432  df-fun 6478  df-fv 6484  df-topon 22780

This theorem is referenced by:  sn0top  22868  indistop  22871  letop  23075  dfac14  23487  cnfldtop  24652  sszcld  24687  iitop  24754  limccnp2  25774  cxpcn3  26639  lmlim  33928  pnfneige0  33932  sxbrsigalem4  34268  poimir  37650  islptre  45616  fourierdlem62  46163