Theorem topontopi 21520

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 21518	. 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
3	1, 2	ax-mp 5	1 ⊢ 𝐽 ∈ Top

Syntax hints:   ∈ wcel 2111  ‘cfv 6324  Topctop 21498  TopOnctopon 21515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-topon 21516

This theorem is referenced by:  sn0top  21604  indistop  21607  letop  21811  dfac14  22223  cnfldtop  23389  sszcld  23422  iitop  23485  limccnp2  24495  cxpcn3  25337  lmlim  31300  pnfneige0  31304  sxbrsigalem4  31655  knoppcnlem10  33954  poimir  35090  islptre  42261  fourierdlem62  42810