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Theorem ispsubclN 38403
Description: The predicate "is a closed projective subspace". (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
psubclset.a 𝐴 = (Atomsβ€˜πΎ)
psubclset.p βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
psubclset.c 𝐢 = (PSubClβ€˜πΎ)
Assertion
Ref Expression
ispsubclN (𝐾 ∈ 𝐷 β†’ (𝑋 ∈ 𝐢 ↔ (𝑋 βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋)))

Proof of Theorem ispsubclN
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 psubclset.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
2 psubclset.p . . . 4 βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
3 psubclset.c . . . 4 𝐢 = (PSubClβ€˜πΎ)
41, 2, 3psubclsetN 38402 . . 3 (𝐾 ∈ 𝐷 β†’ 𝐢 = {π‘₯ ∣ (π‘₯ βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯)})
54eleq2d 2824 . 2 (𝐾 ∈ 𝐷 β†’ (𝑋 ∈ 𝐢 ↔ 𝑋 ∈ {π‘₯ ∣ (π‘₯ βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯)}))
61fvexi 6857 . . . . 5 𝐴 ∈ V
76ssex 5279 . . . 4 (𝑋 βŠ† 𝐴 β†’ 𝑋 ∈ V)
87adantr 482 . . 3 ((𝑋 βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋) β†’ 𝑋 ∈ V)
9 sseq1 3970 . . . 4 (π‘₯ = 𝑋 β†’ (π‘₯ βŠ† 𝐴 ↔ 𝑋 βŠ† 𝐴))
10 2fveq3 6848 . . . . 5 (π‘₯ = 𝑋 β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)))
11 id 22 . . . . 5 (π‘₯ = 𝑋 β†’ π‘₯ = 𝑋)
1210, 11eqeq12d 2753 . . . 4 (π‘₯ = 𝑋 β†’ (( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯ ↔ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋))
139, 12anbi12d 632 . . 3 (π‘₯ = 𝑋 β†’ ((π‘₯ βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯) ↔ (𝑋 βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋)))
148, 13elab3 3639 . 2 (𝑋 ∈ {π‘₯ ∣ (π‘₯ βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯)} ↔ (𝑋 βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋))
155, 14bitrdi 287 1 (𝐾 ∈ 𝐷 β†’ (𝑋 ∈ 𝐢 ↔ (𝑋 βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2714  Vcvv 3446   βŠ† wss 3911  β€˜cfv 6497  Atomscatm 37728  βŠ₯𝑃cpolN 38368  PSubClcpscN 38400
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-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-rab 3409  df-v 3448  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-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-iota 6449  df-fun 6499  df-fv 6505  df-psubclN 38401
This theorem is referenced by:  psubcliN  38404  psubcli2N  38405  0psubclN  38409  1psubclN  38410  atpsubclN  38411  pmapsubclN  38412  ispsubcl2N  38413  osumclN  38433  pexmidN  38435  pexmidlem6N  38441
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