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Theorem ist1-5lem 22423
Description: Lemma for ist1-5 22425 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 22422 . . . . 5 (𝐽 ∈ Kol2 ↔ 𝐽 ≃ (KQ‘𝐽))
31, 2sylib 221 . . . 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 515 . 2 (𝐽𝐴 → (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ 𝐴))
7 hmphsym 22385 . . . . 5 (𝐽 ≃ (KQ‘𝐽) → (KQ‘𝐽) ≃ 𝐽)
82, 7sylbi 220 . . . 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 410 . 2 ((𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ 𝐴) → 𝐽𝐴)
126, 11impbii 212 1 (𝐽𝐴 ↔ (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wcel 2114   class class class wbr 5042  cfv 6334  Kol2ct0 21909  KQckq 22296  chmph 22357
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
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 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-1o 8089  df-map 8395  df-qtop 16771  df-top 21497  df-topon 21514  df-cn 21830  df-t0 21916  df-kq 22297  df-hmeo 22358  df-hmph 22359
This theorem is referenced by:  ist1-5  22425  ishaus3  22426
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