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Theorem toponcom 20713
Description: If 𝐾 is a topology on the base set of topology 𝐽, then 𝐽 is a topology on the base of 𝐾. (Contributed by Mario Carneiro, 22-Aug-2015.)
Assertion
Ref Expression
toponcom ((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOn‘ 𝐽)) → 𝐽 ∈ (TopOn‘ 𝐾))

Proof of Theorem toponcom
StepHypRef Expression
1 toponuni 20700 . . . 4 (𝐾 ∈ (TopOn‘ 𝐽) → 𝐽 = 𝐾)
21eqcomd 2626 . . 3 (𝐾 ∈ (TopOn‘ 𝐽) → 𝐾 = 𝐽)
32anim2i 592 . 2 ((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOn‘ 𝐽)) → (𝐽 ∈ Top ∧ 𝐾 = 𝐽))
4 istopon 20698 . 2 (𝐽 ∈ (TopOn‘ 𝐾) ↔ (𝐽 ∈ Top ∧ 𝐾 = 𝐽))
53, 4sylibr 224 1 ((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOn‘ 𝐽)) → 𝐽 ∈ (TopOn‘ 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988   cuni 4427  cfv 5876  Topctop 20679  TopOnctopon 20696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-topon 20697
This theorem is referenced by:  toponcomb  20714  kgencn3  21342
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