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Theorem tgiun 21582
 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 5823 . . 3 (∀𝑥𝐴 𝐶𝐵 𝑥𝐴 𝐶 = ran (𝑥𝐴𝐶))
21adantl 485 . 2 ((𝐵𝑉 ∧ ∀𝑥𝐴 𝐶𝐵) → 𝑥𝐴 𝐶 = ran (𝑥𝐴𝐶))
3 eqid 2824 . . . 4 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
43rnmptss 6875 . . 3 (∀𝑥𝐴 𝐶𝐵 → ran (𝑥𝐴𝐶) ⊆ 𝐵)
5 eltg3i 21564 . . 3 ((𝐵𝑉 ∧ ran (𝑥𝐴𝐶) ⊆ 𝐵) → ran (𝑥𝐴𝐶) ∈ (topGen‘𝐵))
64, 5sylan2 595 . 2 ((𝐵𝑉 ∧ ∀𝑥𝐴 𝐶𝐵) → ran (𝑥𝐴𝐶) ∈ (topGen‘𝐵))
72, 6eqeltrd 2916 1 ((𝐵𝑉 ∧ ∀𝑥𝐴 𝐶𝐵) → 𝑥𝐴 𝐶 ∈ (topGen‘𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2115  ∀wral 3133   ⊆ wss 3919  ∪ cuni 4825  ∪ ciun 4906   ↦ cmpt 5133  ran crn 5544  ‘cfv 6344  topGenctg 16709 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452 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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-fv 6352  df-topgen 16715 This theorem is referenced by:  txbasval  22209
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