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

Theorem elpmap 39231
Description: Member of a projective map. (Contributed by NM, 27-Jan-2012.)
Hypotheses
Ref Expression
pmapfval.b 𝐡 = (Baseβ€˜πΎ)
pmapfval.l ≀ = (leβ€˜πΎ)
pmapfval.a 𝐴 = (Atomsβ€˜πΎ)
pmapfval.m 𝑀 = (pmapβ€˜πΎ)
Assertion
Ref Expression
elpmap ((𝐾 ∈ 𝐢 ∧ 𝑋 ∈ 𝐡) β†’ (𝑃 ∈ (π‘€β€˜π‘‹) ↔ (𝑃 ∈ 𝐴 ∧ 𝑃 ≀ 𝑋)))

Proof of Theorem elpmap
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 pmapfval.b . . . 4 𝐡 = (Baseβ€˜πΎ)
2 pmapfval.l . . . 4 ≀ = (leβ€˜πΎ)
3 pmapfval.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
4 pmapfval.m . . . 4 𝑀 = (pmapβ€˜πΎ)
51, 2, 3, 4pmapval 39230 . . 3 ((𝐾 ∈ 𝐢 ∧ 𝑋 ∈ 𝐡) β†’ (π‘€β€˜π‘‹) = {π‘₯ ∈ 𝐴 ∣ π‘₯ ≀ 𝑋})
65eleq2d 2815 . 2 ((𝐾 ∈ 𝐢 ∧ 𝑋 ∈ 𝐡) β†’ (𝑃 ∈ (π‘€β€˜π‘‹) ↔ 𝑃 ∈ {π‘₯ ∈ 𝐴 ∣ π‘₯ ≀ 𝑋}))
7 breq1 5151 . . 3 (π‘₯ = 𝑃 β†’ (π‘₯ ≀ 𝑋 ↔ 𝑃 ≀ 𝑋))
87elrab 3682 . 2 (𝑃 ∈ {π‘₯ ∈ 𝐴 ∣ π‘₯ ≀ 𝑋} ↔ (𝑃 ∈ 𝐴 ∧ 𝑃 ≀ 𝑋))
96, 8bitrdi 287 1 ((𝐾 ∈ 𝐢 ∧ 𝑋 ∈ 𝐡) β†’ (𝑃 ∈ (π‘€β€˜π‘‹) ↔ (𝑃 ∈ 𝐴 ∧ 𝑃 ≀ 𝑋)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  {crab 3429   class class class wbr 5148  β€˜cfv 6548  Basecbs 17180  lecple 17240  Atomscatm 38735  pmapcpmap 38970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-pmap 38977
This theorem is referenced by:  pmapjoin  39325  pmapjat1  39326
  Copyright terms: Public domain W3C validator