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Theorem tgdom 14544
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 4224 . 2 (𝐵𝑉 → 𝒫 𝐵 ∈ V)
2 inss1 3393 . . . . 5 (𝐵 ∩ 𝒫 𝑥) ⊆ 𝐵
3 vpwex 4223 . . . . . . 7 𝒫 𝑥 ∈ V
43inex2 4179 . . . . . 6 (𝐵 ∩ 𝒫 𝑥) ∈ V
54elpw 3622 . . . . 5 ((𝐵 ∩ 𝒫 𝑥) ∈ 𝒫 𝐵 ↔ (𝐵 ∩ 𝒫 𝑥) ⊆ 𝐵)
62, 5mpbir 146 . . . 4 (𝐵 ∩ 𝒫 𝑥) ∈ 𝒫 𝐵
76a1i 9 . . 3 (𝑥 ∈ (topGen‘𝐵) → (𝐵 ∩ 𝒫 𝑥) ∈ 𝒫 𝐵)
8 unieq 3859 . . . . . . 7 ((𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦) → (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦))
98adantl 277 . . . . . 6 (((𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) ∧ (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦)) → (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦))
10 eltg4i 14527 . . . . . . 7 (𝑥 ∈ (topGen‘𝐵) → 𝑥 = (𝐵 ∩ 𝒫 𝑥))
1110ad2antrr 488 . . . . . 6 (((𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) ∧ (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦)) → 𝑥 = (𝐵 ∩ 𝒫 𝑥))
12 eltg4i 14527 . . . . . . 7 (𝑦 ∈ (topGen‘𝐵) → 𝑦 = (𝐵 ∩ 𝒫 𝑦))
1312ad2antlr 489 . . . . . 6 (((𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) ∧ (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦)) → 𝑦 = (𝐵 ∩ 𝒫 𝑦))
149, 11, 133eqtr4d 2248 . . . . 5 (((𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) ∧ (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦)) → 𝑥 = 𝑦)
1514ex 115 . . . 4 ((𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) → ((𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦) → 𝑥 = 𝑦))
16 pweq 3619 . . . . 5 (𝑥 = 𝑦 → 𝒫 𝑥 = 𝒫 𝑦)
1716ineq2d 3374 . . . 4 (𝑥 = 𝑦 → (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦))
1815, 17impbid1 142 . . 3 ((𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) → ((𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦) ↔ 𝑥 = 𝑦))
197, 18dom2 6866 . 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 104   = wceq 1373  wcel 2176  Vcvv 2772  cin 3165  wss 3166  𝒫 cpw 3616   cuni 3850   class class class wbr 4044  cfv 5271  cdom 6826  topGenctg 13086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480
This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-dom 6829  df-topgen 13092
This theorem is referenced by: (None)
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