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Theorem tgiun 21515
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 5828 . . 3 (∀𝑥𝐴 𝐶𝐵 𝑥𝐴 𝐶 = ran (𝑥𝐴𝐶))
21adantl 482 . 2 ((𝐵𝑉 ∧ ∀𝑥𝐴 𝐶𝐵) → 𝑥𝐴 𝐶 = ran (𝑥𝐴𝐶))
3 eqid 2818 . . . 4 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
43rnmptss 6878 . . 3 (∀𝑥𝐴 𝐶𝐵 → ran (𝑥𝐴𝐶) ⊆ 𝐵)
5 eltg3i 21497 . . 3 ((𝐵𝑉 ∧ ran (𝑥𝐴𝐶) ⊆ 𝐵) → ran (𝑥𝐴𝐶) ∈ (topGen‘𝐵))
64, 5sylan2 592 . 2 ((𝐵𝑉 ∧ ∀𝑥𝐴 𝐶𝐵) → ran (𝑥𝐴𝐶) ∈ (topGen‘𝐵))
72, 6eqeltrd 2910 1 ((𝐵𝑉 ∧ ∀𝑥𝐴 𝐶𝐵) → 𝑥𝐴 𝐶 ∈ (topGen‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  wss 3933   cuni 4830   ciun 4910  cmpt 5137  ran crn 5549  cfv 6348  topGenctg 16699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-topgen 16705
This theorem is referenced by:  txbasval  22142
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