Theorem elpmap 40633

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	⊢ ((𝐾 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵) → (𝑃 ∈ (𝑀‘𝑋) ↔ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑋)))

Proof of Theorem elpmap

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	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)
5	1, 2, 3, 4	pmapval 40632	. . 3 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) = {𝑥 ∈ 𝐴 ∣ 𝑥 ≤ 𝑋})
6	5	eleq2d 2848	. 2 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵) → (𝑃 ∈ (𝑀‘𝑋) ↔ 𝑃 ∈ {𝑥 ∈ 𝐴 ∣ 𝑥 ≤ 𝑋}))
7		breq1 5110	. . 3 ⊢ (𝑥 = 𝑃 → (𝑥 ≤ 𝑋 ↔ 𝑃 ≤ 𝑋))
8	7	elrab 3648	. 2 ⊢ (𝑃 ∈ {𝑥 ∈ 𝐴 ∣ 𝑥 ≤ 𝑋} ↔ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑋))
9	6, 8	bitrdi 290	1 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵) → (𝑃 ∈ (𝑀‘𝑋) ↔ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑋)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {crab 3414   class class class wbr 5107  ‘cfv 6537  Basecbs 17305  lecple 17353  Atomscatm 40138  pmapcpmap 40372

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-pmap 40379

This theorem is used by:  pmapjoin  40727  pmapjat1  40728