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Theorem ist1-5lem 23844
Description: Lemma for ist1-5 23846 and similar theorems. If 𝐴 is a topological property which implies T0, such as T1 or T2, the property can be "decomposed" into T0 and a non-T0 version of property 𝐴 (which is defined as stating that the Kolmogorov quotient of the space has property 𝐴). For example, if 𝐴 is T1, then the theorem states that a space is T1 iff it is T0 and its Kolmogorov quotient is T1 (we call this property R0). (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypotheses
Ref Expression
ist1-5lem.1 (𝐽𝐴𝐽 ∈ Kol2)
ist1-5lem.2 (𝐽 ≃ (KQ‘𝐽) → (𝐽𝐴 → (KQ‘𝐽) ∈ 𝐴))
ist1-5lem.3 ((KQ‘𝐽) ≃ 𝐽 → ((KQ‘𝐽) ∈ 𝐴𝐽𝐴))
Assertion
Ref Expression
ist1-5lem (𝐽𝐴 ↔ (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ 𝐴))

Proof of Theorem ist1-5lem
StepHypRef Expression
1 ist1-5lem.1 . . 3 (𝐽𝐴𝐽 ∈ Kol2)
2 kqhmph 23843 . . . . 5 (𝐽 ∈ Kol2 ↔ 𝐽 ≃ (KQ‘𝐽))
31, 2sylib 218 . . . 4 (𝐽𝐴𝐽 ≃ (KQ‘𝐽))
4 ist1-5lem.2 . . . 4 (𝐽 ≃ (KQ‘𝐽) → (𝐽𝐴 → (KQ‘𝐽) ∈ 𝐴))
53, 4mpcom 38 . . 3 (𝐽𝐴 → (KQ‘𝐽) ∈ 𝐴)
61, 5jca 511 . 2 (𝐽𝐴 → (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ 𝐴))
7 hmphsym 23806 . . . . 5 (𝐽 ≃ (KQ‘𝐽) → (KQ‘𝐽) ≃ 𝐽)
82, 7sylbi 217 . . . 4 (𝐽 ∈ Kol2 → (KQ‘𝐽) ≃ 𝐽)
9 ist1-5lem.3 . . . 4 ((KQ‘𝐽) ≃ 𝐽 → ((KQ‘𝐽) ∈ 𝐴𝐽𝐴))
108, 9syl 17 . . 3 (𝐽 ∈ Kol2 → ((KQ‘𝐽) ∈ 𝐴𝐽𝐴))
1110imp 406 . 2 ((𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ 𝐴) → 𝐽𝐴)
126, 11impbii 209 1 (𝐽𝐴 ↔ (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2106   class class class wbr 5148  cfv 6563  Kol2ct0 23330  KQckq 23717  chmph 23778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-1o 8505  df-map 8867  df-qtop 17554  df-top 22916  df-topon 22933  df-cn 23251  df-t0 23337  df-kq 23718  df-hmeo 23779  df-hmph 23780
This theorem is referenced by:  ist1-5  23846  ishaus3  23847
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