Theorem elpmap 36896

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 36895	. . 3 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) = {𝑥 ∈ 𝐴 ∣ 𝑥 ≤ 𝑋})
6	5	eleq2d 2900	. 2 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵) → (𝑃 ∈ (𝑀‘𝑋) ↔ 𝑃 ∈ {𝑥 ∈ 𝐴 ∣ 𝑥 ≤ 𝑋}))
7		breq1 5071	. . 3 ⊢ (𝑥 = 𝑃 → (𝑥 ≤ 𝑋 ↔ 𝑃 ≤ 𝑋))
8	7	elrab 3682	. 2 ⊢ (𝑃 ∈ {𝑥 ∈ 𝐴 ∣ 𝑥 ≤ 𝑋} ↔ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑋))
9	6, 8	syl6bb 289	1 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵) → (𝑃 ∈ (𝑀‘𝑋) ↔ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑋)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  {crab 3144   class class class wbr 5068  ‘cfv 6357  Basecbs 16485  lecple 16574  Atomscatm 36401  pmapcpmap 36635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-pmap 36642

This theorem is referenced by:  pmapjoin  36990  pmapjat1  36991