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Theorem t1hmph 21965
Description: T1 is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
Assertion
Ref Expression
t1hmph (𝐽𝐾 → (𝐽 ∈ Fre → 𝐾 ∈ Fre))

Proof of Theorem t1hmph
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 t1top 21505 . 2 (𝐽 ∈ Fre → 𝐽 ∈ Top)
2 cnt1 21525 . 2 ((𝐽 ∈ Fre ∧ 𝑓: 𝐾1-1 𝐽𝑓 ∈ (𝐾 Cn 𝐽)) → 𝐾 ∈ Fre)
31, 2haushmphlem 21961 1 (𝐽𝐾 → (𝐽 ∈ Fre → 𝐾 ∈ Fre))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2166   cuni 4658   class class class wbr 4873  Frect1 21482  chmph 21928
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-1st 7428  df-2nd 7429  df-1o 7826  df-map 8124  df-top 21069  df-topon 21086  df-cld 21194  df-cn 21402  df-t1 21489  df-hmeo 21929  df-hmph 21930
This theorem is referenced by:  t1r0  21995  ist1-5  21996
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