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Theorem ispsubclN 39466
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 39465 . . 3 (𝐾 ∈ 𝐷 β†’ 𝐢 = {π‘₯ ∣ (π‘₯ βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯)})
54eleq2d 2811 . 2 (𝐾 ∈ 𝐷 β†’ (𝑋 ∈ 𝐢 ↔ 𝑋 ∈ {π‘₯ ∣ (π‘₯ βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯)}))
61fvexi 6906 . . . . 5 𝐴 ∈ V
76ssex 5316 . . . 4 (𝑋 βŠ† 𝐴 β†’ 𝑋 ∈ V)
87adantr 479 . . 3 ((𝑋 βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋) β†’ 𝑋 ∈ V)
9 sseq1 3998 . . . 4 (π‘₯ = 𝑋 β†’ (π‘₯ βŠ† 𝐴 ↔ 𝑋 βŠ† 𝐴))
10 2fveq3 6897 . . . . 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 2741 . . . 4 (π‘₯ = 𝑋 β†’ (( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯ ↔ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋))
139, 12anbi12d 630 . . 3 (π‘₯ = 𝑋 β†’ ((π‘₯ βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯) ↔ (𝑋 βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋)))
148, 13elab3 3667 . 2 (𝑋 ∈ {π‘₯ ∣ (π‘₯ βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯)} ↔ (𝑋 βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋))
155, 14bitrdi 286 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 394   = wceq 1533   ∈ wcel 2098  {cab 2702  Vcvv 3463   βŠ† wss 3939  β€˜cfv 6543  Atomscatm 38791  βŠ₯𝑃cpolN 39431  PSubClcpscN 39463
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 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6495  df-fun 6545  df-fv 6551  df-psubclN 39464
This theorem is referenced by:  psubcliN  39467  psubcli2N  39468  0psubclN  39472  1psubclN  39473  atpsubclN  39474  pmapsubclN  39475  ispsubcl2N  39476  osumclN  39496  pexmidN  39498  pexmidlem6N  39504
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