Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elpclN Structured version   Visualization version   GIF version

Theorem elpclN 38358
Description: Membership in the projective subspace closure function. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
pclfval.a 𝐴 = (Atomsβ€˜πΎ)
pclfval.s 𝑆 = (PSubSpβ€˜πΎ)
pclfval.c π‘ˆ = (PClβ€˜πΎ)
elpcl.q 𝑄 ∈ V
Assertion
Ref Expression
elpclN ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† 𝐴) β†’ (𝑄 ∈ (π‘ˆβ€˜π‘‹) ↔ βˆ€π‘¦ ∈ 𝑆 (𝑋 βŠ† 𝑦 β†’ 𝑄 ∈ 𝑦)))
Distinct variable groups:   𝑦,𝐴   𝑦,𝐾   𝑦,𝑆   𝑦,𝑋   𝑦,𝑉   𝑦,𝑄
Allowed substitution hint:   π‘ˆ(𝑦)

Proof of Theorem elpclN
StepHypRef Expression
1 pclfval.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
2 pclfval.s . . . 4 𝑆 = (PSubSpβ€˜πΎ)
3 pclfval.c . . . 4 π‘ˆ = (PClβ€˜πΎ)
41, 2, 3pclvalN 38356 . . 3 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† 𝐴) β†’ (π‘ˆβ€˜π‘‹) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦})
54eleq2d 2824 . 2 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† 𝐴) β†’ (𝑄 ∈ (π‘ˆβ€˜π‘‹) ↔ 𝑄 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦}))
6 elpcl.q . . 3 𝑄 ∈ V
76elintrab 4922 . 2 (𝑄 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦} ↔ βˆ€π‘¦ ∈ 𝑆 (𝑋 βŠ† 𝑦 β†’ 𝑄 ∈ 𝑦))
85, 7bitrdi 287 1 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† 𝐴) β†’ (𝑄 ∈ (π‘ˆβ€˜π‘‹) ↔ βˆ€π‘¦ ∈ 𝑆 (𝑋 βŠ† 𝑦 β†’ 𝑄 ∈ 𝑦)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  {crab 3408  Vcvv 3446   βŠ† wss 3911  βˆ© cint 4908  β€˜cfv 6497  Atomscatm 37728  PSubSpcpsubsp 37962  PClcpclN 38353
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 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  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-int 4909  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-ov 7361  df-psubsp 37969  df-pclN 38354
This theorem is referenced by:  pclfinclN  38416
  Copyright terms: Public domain W3C validator