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Theorem tgiun 21112
Description: The indexed union of a set of basic open sets is in the generated topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
Assertion
Ref Expression
tgiun ((𝐵𝑉 ∧ ∀𝑥𝐴 𝐶𝐵) → 𝑥𝐴 𝐶 ∈ (topGen‘𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑉
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem tgiun
StepHypRef Expression
1 dfiun3g 5582 . . 3 (∀𝑥𝐴 𝐶𝐵 𝑥𝐴 𝐶 = ran (𝑥𝐴𝐶))
21adantl 474 . 2 ((𝐵𝑉 ∧ ∀𝑥𝐴 𝐶𝐵) → 𝑥𝐴 𝐶 = ran (𝑥𝐴𝐶))
3 eqid 2799 . . . 4 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
43rnmptss 6618 . . 3 (∀𝑥𝐴 𝐶𝐵 → ran (𝑥𝐴𝐶) ⊆ 𝐵)
5 eltg3i 21094 . . 3 ((𝐵𝑉 ∧ ran (𝑥𝐴𝐶) ⊆ 𝐵) → ran (𝑥𝐴𝐶) ∈ (topGen‘𝐵))
64, 5sylan2 587 . 2 ((𝐵𝑉 ∧ ∀𝑥𝐴 𝐶𝐵) → ran (𝑥𝐴𝐶) ∈ (topGen‘𝐵))
72, 6eqeltrd 2878 1 ((𝐵𝑉 ∧ ∀𝑥𝐴 𝐶𝐵) → 𝑥𝐴 𝐶 ∈ (topGen‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  wral 3089  wss 3769   cuni 4628   ciun 4710  cmpt 4922  ran crn 5313  cfv 6101  topGenctg 16413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-topgen 16419
This theorem is referenced by:  txbasval  21738
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