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Theorem tgdom 22825
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 (𝐡 ∈ 𝑉 β†’ (topGenβ€˜π΅) β‰Ό 𝒫 𝐡)

Proof of Theorem tgdom
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwexg 5367 . 2 (𝐡 ∈ 𝑉 β†’ 𝒫 𝐡 ∈ V)
2 inss1 4221 . . . . 5 (𝐡 ∩ 𝒫 π‘₯) βŠ† 𝐡
3 vpwex 5366 . . . . . . 7 𝒫 π‘₯ ∈ V
43inex2 5309 . . . . . 6 (𝐡 ∩ 𝒫 π‘₯) ∈ V
54elpw 4599 . . . . 5 ((𝐡 ∩ 𝒫 π‘₯) ∈ 𝒫 𝐡 ↔ (𝐡 ∩ 𝒫 π‘₯) βŠ† 𝐡)
62, 5mpbir 230 . . . 4 (𝐡 ∩ 𝒫 π‘₯) ∈ 𝒫 𝐡
76a1i 11 . . 3 (π‘₯ ∈ (topGenβ€˜π΅) β†’ (𝐡 ∩ 𝒫 π‘₯) ∈ 𝒫 𝐡)
8 unieq 4911 . . . . . . 7 ((𝐡 ∩ 𝒫 π‘₯) = (𝐡 ∩ 𝒫 𝑦) β†’ βˆͺ (𝐡 ∩ 𝒫 π‘₯) = βˆͺ (𝐡 ∩ 𝒫 𝑦))
98adantl 481 . . . . . 6 (((π‘₯ ∈ (topGenβ€˜π΅) ∧ 𝑦 ∈ (topGenβ€˜π΅)) ∧ (𝐡 ∩ 𝒫 π‘₯) = (𝐡 ∩ 𝒫 𝑦)) β†’ βˆͺ (𝐡 ∩ 𝒫 π‘₯) = βˆͺ (𝐡 ∩ 𝒫 𝑦))
10 eltg4i 22807 . . . . . . 7 (π‘₯ ∈ (topGenβ€˜π΅) β†’ π‘₯ = βˆͺ (𝐡 ∩ 𝒫 π‘₯))
1110ad2antrr 723 . . . . . 6 (((π‘₯ ∈ (topGenβ€˜π΅) ∧ 𝑦 ∈ (topGenβ€˜π΅)) ∧ (𝐡 ∩ 𝒫 π‘₯) = (𝐡 ∩ 𝒫 𝑦)) β†’ π‘₯ = βˆͺ (𝐡 ∩ 𝒫 π‘₯))
12 eltg4i 22807 . . . . . . 7 (𝑦 ∈ (topGenβ€˜π΅) β†’ 𝑦 = βˆͺ (𝐡 ∩ 𝒫 𝑦))
1312ad2antlr 724 . . . . . 6 (((π‘₯ ∈ (topGenβ€˜π΅) ∧ 𝑦 ∈ (topGenβ€˜π΅)) ∧ (𝐡 ∩ 𝒫 π‘₯) = (𝐡 ∩ 𝒫 𝑦)) β†’ 𝑦 = βˆͺ (𝐡 ∩ 𝒫 𝑦))
149, 11, 133eqtr4d 2774 . . . . 5 (((π‘₯ ∈ (topGenβ€˜π΅) ∧ 𝑦 ∈ (topGenβ€˜π΅)) ∧ (𝐡 ∩ 𝒫 π‘₯) = (𝐡 ∩ 𝒫 𝑦)) β†’ π‘₯ = 𝑦)
1514ex 412 . . . 4 ((π‘₯ ∈ (topGenβ€˜π΅) ∧ 𝑦 ∈ (topGenβ€˜π΅)) β†’ ((𝐡 ∩ 𝒫 π‘₯) = (𝐡 ∩ 𝒫 𝑦) β†’ π‘₯ = 𝑦))
16 pweq 4609 . . . . 5 (π‘₯ = 𝑦 β†’ 𝒫 π‘₯ = 𝒫 𝑦)
1716ineq2d 4205 . . . 4 (π‘₯ = 𝑦 β†’ (𝐡 ∩ 𝒫 π‘₯) = (𝐡 ∩ 𝒫 𝑦))
1815, 17impbid1 224 . . 3 ((π‘₯ ∈ (topGenβ€˜π΅) ∧ 𝑦 ∈ (topGenβ€˜π΅)) β†’ ((𝐡 ∩ 𝒫 π‘₯) = (𝐡 ∩ 𝒫 𝑦) ↔ π‘₯ = 𝑦))
197, 18dom2 8988 . 2 (𝒫 𝐡 ∈ V β†’ (topGenβ€˜π΅) β‰Ό 𝒫 𝐡)
201, 19syl 17 1 (𝐡 ∈ 𝑉 β†’ (topGenβ€˜π΅) β‰Ό 𝒫 𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3466   ∩ cin 3940   βŠ† wss 3941  π’« cpw 4595  βˆͺ cuni 4900   class class class wbr 5139  β€˜cfv 6534   β‰Ό cdom 8934  topGenctg 17388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-dom 8938  df-topgen 17394
This theorem is referenced by:  2ndcredom  23298  kelac2lem  42358
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