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Theorem elpmap 35559
 Description: Member of a projective map. (Contributed by NM, 27-Jan-2012.)
Hypotheses
Ref Expression
pmapfval.b 𝐵 = (Base‘𝐾)
pmapfval.l = (le‘𝐾)
pmapfval.a 𝐴 = (Atoms‘𝐾)
pmapfval.m 𝑀 = (pmap‘𝐾)
Assertion
Ref Expression
elpmap ((𝐾𝐶𝑋𝐵) → (𝑃 ∈ (𝑀𝑋) ↔ (𝑃𝐴𝑃 𝑋)))

Proof of Theorem elpmap
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 pmapfval.b . . . 4 𝐵 = (Base‘𝐾)
2 pmapfval.l . . . 4 = (le‘𝐾)
3 pmapfval.a . . . 4 𝐴 = (Atoms‘𝐾)
4 pmapfval.m . . . 4 𝑀 = (pmap‘𝐾)
51, 2, 3, 4pmapval 35558 . . 3 ((𝐾𝐶𝑋𝐵) → (𝑀𝑋) = {𝑥𝐴𝑥 𝑋})
65eleq2d 2835 . 2 ((𝐾𝐶𝑋𝐵) → (𝑃 ∈ (𝑀𝑋) ↔ 𝑃 ∈ {𝑥𝐴𝑥 𝑋}))
7 breq1 4787 . . 3 (𝑥 = 𝑃 → (𝑥 𝑋𝑃 𝑋))
87elrab 3513 . 2 (𝑃 ∈ {𝑥𝐴𝑥 𝑋} ↔ (𝑃𝐴𝑃 𝑋))
96, 8syl6bb 276 1 ((𝐾𝐶𝑋𝐵) → (𝑃 ∈ (𝑀𝑋) ↔ (𝑃𝐴𝑃 𝑋)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1630   ∈ wcel 2144  {crab 3064   class class class wbr 4784  ‘cfv 6031  Basecbs 16063  lecple 16155  Atomscatm 35065  pmapcpmap 35298 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pr 5034 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-pmap 35305 This theorem is referenced by:  pmapjoin  35653  pmapjat1  35654
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