MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tgiun Structured version   Visualization version   GIF version

Theorem tgiun 21517
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 5829 . . 3 (∀𝑥𝐴 𝐶𝐵 𝑥𝐴 𝐶 = ran (𝑥𝐴𝐶))
21adantl 482 . 2 ((𝐵𝑉 ∧ ∀𝑥𝐴 𝐶𝐵) → 𝑥𝐴 𝐶 = ran (𝑥𝐴𝐶))
3 eqid 2821 . . . 4 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
43rnmptss 6879 . . 3 (∀𝑥𝐴 𝐶𝐵 → ran (𝑥𝐴𝐶) ⊆ 𝐵)
5 eltg3i 21499 . . 3 ((𝐵𝑉 ∧ ran (𝑥𝐴𝐶) ⊆ 𝐵) → ran (𝑥𝐴𝐶) ∈ (topGen‘𝐵))
64, 5sylan2 592 . 2 ((𝐵𝑉 ∧ ∀𝑥𝐴 𝐶𝐵) → ran (𝑥𝐴𝐶) ∈ (topGen‘𝐵))
72, 6eqeltrd 2913 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 3138  wss 3935   cuni 4832   ciun 4912  cmpt 5138  ran crn 5550  cfv 6349  topGenctg 16701
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 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  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 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-fv 6357  df-topgen 16707
This theorem is referenced by:  txbasval  22144
  Copyright terms: Public domain W3C validator