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Theorem toponcomb 13980
Description: Biconditional form of toponcom 13979. (Contributed by BJ, 5-Dec-2021.)
Assertion
Ref Expression
toponcomb ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∈ (TopOn‘ 𝐾) ↔ 𝐾 ∈ (TopOn‘ 𝐽)))

Proof of Theorem toponcomb
StepHypRef Expression
1 toponcom 13979 . . . 4 ((𝐾 ∈ Top ∧ 𝐽 ∈ (TopOn‘ 𝐾)) → 𝐾 ∈ (TopOn‘ 𝐽))
21ex 115 . . 3 (𝐾 ∈ Top → (𝐽 ∈ (TopOn‘ 𝐾) → 𝐾 ∈ (TopOn‘ 𝐽)))
32adantl 277 . 2 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∈ (TopOn‘ 𝐾) → 𝐾 ∈ (TopOn‘ 𝐽)))
4 toponcom 13979 . . . 4 ((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOn‘ 𝐽)) → 𝐽 ∈ (TopOn‘ 𝐾))
54ex 115 . . 3 (𝐽 ∈ Top → (𝐾 ∈ (TopOn‘ 𝐽) → 𝐽 ∈ (TopOn‘ 𝐾)))
65adantr 276 . 2 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐾 ∈ (TopOn‘ 𝐽) → 𝐽 ∈ (TopOn‘ 𝐾)))
73, 6impbid 129 1 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∈ (TopOn‘ 𝐾) ↔ 𝐾 ∈ (TopOn‘ 𝐽)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wcel 2160   cuni 3824  cfv 5235  Topctop 13949  TopOnctopon 13962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-topon 13963
This theorem is referenced by: (None)
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