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Theorem t1hmph 21817
 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 21357 . 2 (𝐽 ∈ Fre → 𝐽 ∈ Top)
2 cnt1 21377 . 2 ((𝐽 ∈ Fre ∧ 𝑓: 𝐾1-1 𝐽𝑓 ∈ (𝐾 Cn 𝐽)) → 𝐾 ∈ Fre)
31, 2haushmphlem 21813 1 (𝐽𝐾 → (𝐽 ∈ Fre → 𝐾 ∈ Fre))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2140  ∪ cuni 4589   class class class wbr 4805  Frect1 21334   ≃ chmph 21780 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-suc 5891  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-1st 7335  df-2nd 7336  df-1o 7731  df-map 8028  df-top 20922  df-topon 20939  df-cld 21046  df-cn 21254  df-t1 21341  df-hmeo 21781  df-hmph 21782 This theorem is referenced by:  t1r0  21847  ist1-5  21848
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