Theorem dibelval1st1 41117

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

Hypotheses

Ref	Expression
dibelval1st1.b	⊢ 𝐵 = (Base‘𝐾)
dibelval1st1.l	⊢ ≤ = (le‘𝐾)
dibelval1st1.h	⊢ 𝐻 = (LHyp‘𝐾)
dibelval1st1.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dibelval1st1.i	⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)

Assertion

Ref	Expression
dibelval1st1	⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑋)) → (1st ‘𝑌) ∈ 𝑇)

Proof of Theorem dibelval1st1

Step	Hyp	Ref	Expression
1		dibelval1st1.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		dibelval1st1.l	. . 3 ⊢ ≤ = (le‘𝐾)
3		dibelval1st1.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
4		eqid 2729	. . 3 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
5		dibelval1st1.i	. . 3 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)
6	1, 2, 3, 4, 5	dibelval1st 41116	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑋)) → (1st ‘𝑌) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋))
7		dibelval1st1.t	. . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
8	1, 2, 3, 7, 4	diael 41010	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (1st ‘𝑌) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋)) → (1st ‘𝑌) ∈ 𝑇)
9	6, 8	syld3an3 1411	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑋)) → (1st ‘𝑌) ∈ 𝑇)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   class class class wbr 5102  ‘cfv 6499  1^st c1st 7945  Basecbs 17155  lecple 17203  LHypclh 39951  LTrncltrn 40068  DIsoAcdia 40995  DIsoBcdib 41105

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-1st 7947  df-disoa 40996  df-dib 41106

This theorem is referenced by:  diblss  41137