Theorem dibelval1st 41129

Description: Membership in value of the partial isomorphism B for a lattice 𝐾. (Contributed by NM, 13-Feb-2014.)

Hypotheses

Ref	Expression
dibelval1.b	⊢ 𝐵 = (Base‘𝐾)
dibelval1.l	⊢ ≤ = (le‘𝐾)
dibelval1.h	⊢ 𝐻 = (LHyp‘𝐾)
dibelval1.j	⊢ 𝐽 = ((DIsoA‘𝐾)‘𝑊)
dibelval1.i	⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)

Assertion

Ref	Expression
dibelval1st	⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑋)) → (1st ‘𝑌) ∈ (𝐽‘𝑋))

Proof of Theorem dibelval1st

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dibelval1.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
2		dibelval1.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
3		dibelval1.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4		eqid 2736	. . . . 5 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
5		eqid 2736	. . . . 5 ⊢ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))
6		dibelval1.j	. . . . 5 ⊢ 𝐽 = ((DIsoA‘𝐾)‘𝑊)
7		dibelval1.i	. . . . 5 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)
8	1, 2, 3, 4, 5, 6, 7	dibval2 41124	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = ((𝐽‘𝑋) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))}))
9	8	eleq2d 2826	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑋) ↔ 𝑌 ∈ ((𝐽‘𝑋) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))})))
10	9	biimp3a 1471	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑋)) → 𝑌 ∈ ((𝐽‘𝑋) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))}))
11		xp1st 8042	. 2 ⊢ (𝑌 ∈ ((𝐽‘𝑋) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))}) → (1st ‘𝑌) ∈ (𝐽‘𝑋))
12	10, 11	syl 17	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑋)) → (1st ‘𝑌) ∈ (𝐽‘𝑋))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  {csn 4624   class class class wbr 5141   ↦ cmpt 5223   I cid 5575   × cxp 5681   ↾ cres 5685  ‘cfv 6559  1^st c1st 8008  Basecbs 17243  lecple 17300  LHypclh 39964  LTrncltrn 40081  DIsoAcdia 41008  DIsoBcdib 41118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5277  ax-sep 5294  ax-nul 5304  ax-pow 5363  ax-pr 5430  ax-un 7751

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566  df-fv 6567  df-1st 8010  df-disoa 41009  df-dib 41119

This theorem is referenced by:  dibelval1st1  41130  dibelval1st2N  41131  diblss  41150