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Theorem 2basgeng 12033
Description: Conditions that determine the equality of two generated topologies. (Contributed by NM, 8-May-2007.) (Revised by Jim Kingdon, 5-Mar-2023.)
Assertion
Ref Expression
2basgeng ((𝐵𝑉𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → (topGen‘𝐵) = (topGen‘𝐶))

Proof of Theorem 2basgeng
StepHypRef Expression
1 tgvalex 12001 . . . . 5 (𝐵𝑉 → (topGen‘𝐵) ∈ V)
213ad2ant1 970 . . . 4 ((𝐵𝑉𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → (topGen‘𝐵) ∈ V)
3 simp3 951 . . . 4 ((𝐵𝑉𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → 𝐶 ⊆ (topGen‘𝐵))
42, 3ssexd 4008 . . 3 ((𝐵𝑉𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → 𝐶 ∈ V)
5 simp2 950 . . 3 ((𝐵𝑉𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → 𝐵𝐶)
6 tgss 12014 . . 3 ((𝐶 ∈ V ∧ 𝐵𝐶) → (topGen‘𝐵) ⊆ (topGen‘𝐶))
74, 5, 6syl2anc 406 . 2 ((𝐵𝑉𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → (topGen‘𝐵) ⊆ (topGen‘𝐶))
8 simp1 949 . . . 4 ((𝐵𝑉𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → 𝐵𝑉)
9 tgss3 12029 . . . 4 ((𝐶 ∈ V ∧ 𝐵𝑉) → ((topGen‘𝐶) ⊆ (topGen‘𝐵) ↔ 𝐶 ⊆ (topGen‘𝐵)))
104, 8, 9syl2anc 406 . . 3 ((𝐵𝑉𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → ((topGen‘𝐶) ⊆ (topGen‘𝐵) ↔ 𝐶 ⊆ (topGen‘𝐵)))
113, 10mpbird 166 . 2 ((𝐵𝑉𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → (topGen‘𝐶) ⊆ (topGen‘𝐵))
127, 11eqssd 3064 1 ((𝐵𝑉𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → (topGen‘𝐵) = (topGen‘𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  w3a 930   = wceq 1299  wcel 1448  Vcvv 2641  wss 3021  cfv 5059  topGenctg 11917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-sbc 2863  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-iota 5024  df-fun 5061  df-fv 5067  df-topgen 11923
This theorem is referenced by:  txbasval  12217  tgioo  12465  tgqioo  12466
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