Theorem topontopi 22942

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 22940	. 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
3	1, 2	ax-mp 5	1 ⊢ 𝐽 ∈ Top

Syntax hints:   ∈ wcel 2108  ‘cfv 6573  Topctop 22920  TopOnctopon 22937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-topon 22938

This theorem is referenced by:  sn0top  23027  indistop  23030  letop  23235  dfac14  23647  cnfldtop  24825  sszcld  24858  iitop  24925  limccnp2  25947  cxpcn3  26809  lmlim  33893  pnfneige0  33897  sxbrsigalem4  34252  poimir  37613  islptre  45540  fourierdlem62  46089