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Theorem tg1 21567
Description: Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)
Assertion
Ref Expression
tg1 (𝐴 ∈ (topGen‘𝐵) → 𝐴 𝐵)

Proof of Theorem tg1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvdm 6684 . 2 (𝐴 ∈ (topGen‘𝐵) → 𝐵 ∈ dom topGen)
2 eltg2 21561 . . 3 (𝐵 ∈ dom topGen → (𝐴 ∈ (topGen‘𝐵) ↔ (𝐴 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝑦𝐴))))
32simprbda 502 . 2 ((𝐵 ∈ dom topGen ∧ 𝐴 ∈ (topGen‘𝐵)) → 𝐴 𝐵)
41, 3mpancom 687 1 (𝐴 ∈ (topGen‘𝐵) → 𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2114  wral 3130  wrex 3131  wss 3908   cuni 4813  dom cdm 5532  cfv 6334  topGenctg 16702
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-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-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  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-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-iota 6293  df-fun 6336  df-fv 6342  df-topgen 16708
This theorem is referenced by:  unitg  21570  tgcl  21572  ontgval  33853
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