MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  t0hmph Structured version   Visualization version   GIF version

Theorem t0hmph 22374
Description: T0 is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
Assertion
Ref Expression
t0hmph (𝐽𝐾 → (𝐽 ∈ Kol2 → 𝐾 ∈ Kol2))

Proof of Theorem t0hmph
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 t0top 21913 . 2 (𝐽 ∈ Kol2 → 𝐽 ∈ Top)
2 cnt0 21930 . 2 ((𝐽 ∈ Kol2 ∧ 𝑓: 𝐾1-1 𝐽𝑓 ∈ (𝐾 Cn 𝐽)) → 𝐾 ∈ Kol2)
31, 2haushmphlem 22371 1 (𝐽𝐾 → (𝐽 ∈ Kol2 → 𝐾 ∈ Kol2))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2115   cuni 4811   class class class wbr 5039  Kol2ct0 21890  chmph 22338
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
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 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-1st 7664  df-2nd 7665  df-1o 8077  df-map 8383  df-top 21478  df-topon 21495  df-cn 21811  df-t0 21897  df-hmeo 22339  df-hmph 22340
This theorem is referenced by:  t0kq  22402  kqhmph  22403
  Copyright terms: Public domain W3C validator