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Theorem tgdom 12241
Description: A space has no more open sets than subsets of a basis. (Contributed by Stefan O'Rear, 22-Feb-2015.) (Revised by Mario Carneiro, 9-Apr-2015.)
Assertion
Ref Expression
tgdom (𝐵𝑉 → (topGen‘𝐵) ≼ 𝒫 𝐵)

Proof of Theorem tgdom
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwexg 4104 . 2 (𝐵𝑉 → 𝒫 𝐵 ∈ V)
2 inss1 3296 . . . . 5 (𝐵 ∩ 𝒫 𝑥) ⊆ 𝐵
3 vpwex 4103 . . . . . . 7 𝒫 𝑥 ∈ V
43inex2 4063 . . . . . 6 (𝐵 ∩ 𝒫 𝑥) ∈ V
54elpw 3516 . . . . 5 ((𝐵 ∩ 𝒫 𝑥) ∈ 𝒫 𝐵 ↔ (𝐵 ∩ 𝒫 𝑥) ⊆ 𝐵)
62, 5mpbir 145 . . . 4 (𝐵 ∩ 𝒫 𝑥) ∈ 𝒫 𝐵
76a1i 9 . . 3 (𝑥 ∈ (topGen‘𝐵) → (𝐵 ∩ 𝒫 𝑥) ∈ 𝒫 𝐵)
8 unieq 3745 . . . . . . 7 ((𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦) → (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦))
98adantl 275 . . . . . 6 (((𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) ∧ (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦)) → (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦))
10 eltg4i 12224 . . . . . . 7 (𝑥 ∈ (topGen‘𝐵) → 𝑥 = (𝐵 ∩ 𝒫 𝑥))
1110ad2antrr 479 . . . . . 6 (((𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) ∧ (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦)) → 𝑥 = (𝐵 ∩ 𝒫 𝑥))
12 eltg4i 12224 . . . . . . 7 (𝑦 ∈ (topGen‘𝐵) → 𝑦 = (𝐵 ∩ 𝒫 𝑦))
1312ad2antlr 480 . . . . . 6 (((𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) ∧ (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦)) → 𝑦 = (𝐵 ∩ 𝒫 𝑦))
149, 11, 133eqtr4d 2182 . . . . 5 (((𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) ∧ (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦)) → 𝑥 = 𝑦)
1514ex 114 . . . 4 ((𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) → ((𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦) → 𝑥 = 𝑦))
16 pweq 3513 . . . . 5 (𝑥 = 𝑦 → 𝒫 𝑥 = 𝒫 𝑦)
1716ineq2d 3277 . . . 4 (𝑥 = 𝑦 → (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦))
1815, 17impbid1 141 . . 3 ((𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) → ((𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦) ↔ 𝑥 = 𝑦))
197, 18dom2 6669 . 2 (𝒫 𝐵 ∈ V → (topGen‘𝐵) ≼ 𝒫 𝐵)
201, 19syl 14 1 (𝐵𝑉 → (topGen‘𝐵) ≼ 𝒫 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1331  wcel 1480  Vcvv 2686  cin 3070  wss 3071  𝒫 cpw 3510   cuni 3736   class class class wbr 3929  cfv 5123  cdom 6633  topGenctg 12135
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-dom 6636  df-topgen 12141
This theorem is referenced by: (None)
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