Theorem dibelval1st1 41649

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 2740	. . 3 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
5		dibelval1st1.i	. . 3 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)
6	1, 2, 3, 4, 5	dibelval1st 41648	. 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 41542	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (1st ‘𝑌) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋)) → (1st ‘𝑌) ∈ 𝑇)
9	6, 8	syld3an3 1417	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑋)) → (1st ‘𝑌) ∈ 𝑇)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   class class class wbr 5079  ‘cfv 6492  1^st c1st 7936  Basecbs 17177  lecple 17225  LHypclh 40483  LTrncltrn 40600  DIsoAcdia 41527  DIsoBcdib 41637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-1st 7938  df-disoa 41528  df-dib 41638

This theorem is referenced by:  diblss  41669