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Theorem 2basgen 22998
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 6918 . . . 4 (topGen‘𝐵) ∈ V
21ssex 5320 . . 3 (𝐶 ⊆ (topGen‘𝐵) → 𝐶 ∈ V)
3 simpl 482 . . 3 ((𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → 𝐵𝐶)
4 tgss 22976 . . 3 ((𝐶 ∈ V ∧ 𝐵𝐶) → (topGen‘𝐵) ⊆ (topGen‘𝐶))
52, 3, 4syl2an2 686 . 2 ((𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → (topGen‘𝐵) ⊆ (topGen‘𝐶))
6 simpr 484 . . 3 ((𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → 𝐶 ⊆ (topGen‘𝐵))
7 ssexg 5322 . . . . 5 ((𝐵𝐶𝐶 ∈ V) → 𝐵 ∈ V)
82, 7sylan2 593 . . . 4 ((𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → 𝐵 ∈ V)
9 tgss3 22994 . . . 4 ((𝐶 ∈ V ∧ 𝐵 ∈ V) → ((topGen‘𝐶) ⊆ (topGen‘𝐵) ↔ 𝐶 ⊆ (topGen‘𝐵)))
102, 8, 9syl2an2 686 . . 3 ((𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → ((topGen‘𝐶) ⊆ (topGen‘𝐵) ↔ 𝐶 ⊆ (topGen‘𝐵)))
116, 10mpbird 257 . 2 ((𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → (topGen‘𝐶) ⊆ (topGen‘𝐵))
125, 11eqssd 4000 1 ((𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → (topGen‘𝐵) = (topGen‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  Vcvv 3479  wss 3950  cfv 6560  topGenctg 17483
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-topgen 17489
This theorem is referenced by:  leordtval2  23221  2ndcsb  23458  txbasval  23615  prdsxmslem2  24543  tgioo  24818  tgqioo  24822
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