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Theorem 2basgen 20605
Description: Conditions that determine the equality of two generated topologies. (Contributed by NM, 8-May-2007.) (Revised by Mario Carneiro, 2-Sep-2015.)
Assertion
Ref Expression
2basgen ((𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → (topGen‘𝐵) = (topGen‘𝐶))

Proof of Theorem 2basgen
StepHypRef Expression
1 fvex 6113 . . . . 5 (topGen‘𝐵) ∈ V
21ssex 4730 . . . 4 (𝐶 ⊆ (topGen‘𝐵) → 𝐶 ∈ V)
32adantl 481 . . 3 ((𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → 𝐶 ∈ V)
4 simpl 472 . . 3 ((𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → 𝐵𝐶)
5 tgss 20583 . . 3 ((𝐶 ∈ V ∧ 𝐵𝐶) → (topGen‘𝐵) ⊆ (topGen‘𝐶))
63, 4, 5syl2anc 691 . 2 ((𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → (topGen‘𝐵) ⊆ (topGen‘𝐶))
7 simpr 476 . . 3 ((𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → 𝐶 ⊆ (topGen‘𝐵))
8 ssexg 4732 . . . . 5 ((𝐵𝐶𝐶 ∈ V) → 𝐵 ∈ V)
92, 8sylan2 490 . . . 4 ((𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → 𝐵 ∈ V)
10 tgss3 20601 . . . 4 ((𝐶 ∈ V ∧ 𝐵 ∈ V) → ((topGen‘𝐶) ⊆ (topGen‘𝐵) ↔ 𝐶 ⊆ (topGen‘𝐵)))
113, 9, 10syl2anc 691 . . 3 ((𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → ((topGen‘𝐶) ⊆ (topGen‘𝐵) ↔ 𝐶 ⊆ (topGen‘𝐵)))
127, 11mpbird 246 . 2 ((𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → (topGen‘𝐶) ⊆ (topGen‘𝐵))
136, 12eqssd 3585 1 ((𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → (topGen‘𝐵) = (topGen‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  Vcvv 3173  wss 3540  cfv 5804  topGenctg 15921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-topgen 15927
This theorem is referenced by:  leordtval2  20826  2ndcsb  21062  txbasval  21219  prdsxmslem2  22144  tgioo  22407  tgqioo  22411
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