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Theorem tgdom 22344
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 5334 . 2 (𝐡 ∈ 𝑉 β†’ 𝒫 𝐡 ∈ V)
2 inss1 4189 . . . . 5 (𝐡 ∩ 𝒫 π‘₯) βŠ† 𝐡
3 vpwex 5333 . . . . . . 7 𝒫 π‘₯ ∈ V
43inex2 5276 . . . . . 6 (𝐡 ∩ 𝒫 π‘₯) ∈ V
54elpw 4565 . . . . 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 4877 . . . . . . 7 ((𝐡 ∩ 𝒫 π‘₯) = (𝐡 ∩ 𝒫 𝑦) β†’ βˆͺ (𝐡 ∩ 𝒫 π‘₯) = βˆͺ (𝐡 ∩ 𝒫 𝑦))
98adantl 483 . . . . . 6 (((π‘₯ ∈ (topGenβ€˜π΅) ∧ 𝑦 ∈ (topGenβ€˜π΅)) ∧ (𝐡 ∩ 𝒫 π‘₯) = (𝐡 ∩ 𝒫 𝑦)) β†’ βˆͺ (𝐡 ∩ 𝒫 π‘₯) = βˆͺ (𝐡 ∩ 𝒫 𝑦))
10 eltg4i 22326 . . . . . . 7 (π‘₯ ∈ (topGenβ€˜π΅) β†’ π‘₯ = βˆͺ (𝐡 ∩ 𝒫 π‘₯))
1110ad2antrr 725 . . . . . 6 (((π‘₯ ∈ (topGenβ€˜π΅) ∧ 𝑦 ∈ (topGenβ€˜π΅)) ∧ (𝐡 ∩ 𝒫 π‘₯) = (𝐡 ∩ 𝒫 𝑦)) β†’ π‘₯ = βˆͺ (𝐡 ∩ 𝒫 π‘₯))
12 eltg4i 22326 . . . . . . 7 (𝑦 ∈ (topGenβ€˜π΅) β†’ 𝑦 = βˆͺ (𝐡 ∩ 𝒫 𝑦))
1312ad2antlr 726 . . . . . 6 (((π‘₯ ∈ (topGenβ€˜π΅) ∧ 𝑦 ∈ (topGenβ€˜π΅)) ∧ (𝐡 ∩ 𝒫 π‘₯) = (𝐡 ∩ 𝒫 𝑦)) β†’ 𝑦 = βˆͺ (𝐡 ∩ 𝒫 𝑦))
149, 11, 133eqtr4d 2783 . . . . 5 (((π‘₯ ∈ (topGenβ€˜π΅) ∧ 𝑦 ∈ (topGenβ€˜π΅)) ∧ (𝐡 ∩ 𝒫 π‘₯) = (𝐡 ∩ 𝒫 𝑦)) β†’ π‘₯ = 𝑦)
1514ex 414 . . . 4 ((π‘₯ ∈ (topGenβ€˜π΅) ∧ 𝑦 ∈ (topGenβ€˜π΅)) β†’ ((𝐡 ∩ 𝒫 π‘₯) = (𝐡 ∩ 𝒫 𝑦) β†’ π‘₯ = 𝑦))
16 pweq 4575 . . . . 5 (π‘₯ = 𝑦 β†’ 𝒫 π‘₯ = 𝒫 𝑦)
1716ineq2d 4173 . . . 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 8938 . 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 397   = wceq 1542   ∈ wcel 2107  Vcvv 3444   ∩ cin 3910   βŠ† wss 3911  π’« cpw 4561  βˆͺ cuni 4866   class class class wbr 5106  β€˜cfv 6497   β‰Ό cdom 8884  topGenctg 17324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-dom 8888  df-topgen 17330
This theorem is referenced by:  2ndcredom  22817  kelac2lem  41434
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