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Theorem 2basgeng 12835
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 12803 . . . . 5 (𝐵𝑉 → (topGen‘𝐵) ∈ V)
213ad2ant1 1013 . . . 4 ((𝐵𝑉𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → (topGen‘𝐵) ∈ V)
3 simp3 994 . . . 4 ((𝐵𝑉𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → 𝐶 ⊆ (topGen‘𝐵))
42, 3ssexd 4127 . . 3 ((𝐵𝑉𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → 𝐶 ∈ V)
5 simp2 993 . . 3 ((𝐵𝑉𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → 𝐵𝐶)
6 tgss 12816 . . 3 ((𝐶 ∈ V ∧ 𝐵𝐶) → (topGen‘𝐵) ⊆ (topGen‘𝐶))
74, 5, 6syl2anc 409 . 2 ((𝐵𝑉𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → (topGen‘𝐵) ⊆ (topGen‘𝐶))
8 simp1 992 . . . 4 ((𝐵𝑉𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → 𝐵𝑉)
9 tgss3 12831 . . . 4 ((𝐶 ∈ V ∧ 𝐵𝑉) → ((topGen‘𝐶) ⊆ (topGen‘𝐵) ↔ 𝐶 ⊆ (topGen‘𝐵)))
104, 8, 9syl2anc 409 . . 3 ((𝐵𝑉𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → ((topGen‘𝐶) ⊆ (topGen‘𝐵) ↔ 𝐶 ⊆ (topGen‘𝐵)))
113, 10mpbird 166 . 2 ((𝐵𝑉𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → (topGen‘𝐶) ⊆ (topGen‘𝐵))
127, 11eqssd 3164 1 ((𝐵𝑉𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → (topGen‘𝐵) = (topGen‘𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  w3a 973   = wceq 1348  wcel 2141  Vcvv 2730  wss 3121  cfv 5196  topGenctg 12581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-iota 5158  df-fun 5198  df-fv 5204  df-topgen 12587
This theorem is referenced by:  txbasval  13020  tgioo  13299  tgqioo  13300
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