Theorem dihopelvalbN 40715

Description: Ordered pair member of the partial isomorphism H for argument under 𝑊. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dihval3.b	⊢ 𝐵 = (Base‘𝐾)
dihval3.l	⊢ ≤ = (le‘𝐾)
dihval3.h	⊢ 𝐻 = (LHyp‘𝐾)
dihval3.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dihval3.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
dihval3.o	⊢ 𝑂 = (𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵))
dihval3.i	⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)

Assertion

Ref	Expression
dihopelvalbN	⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼‘𝑋) ↔ ((𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑋) ∧ 𝑆 = 𝑂)))

Distinct variable groups:   𝑔,𝐾   𝑇,𝑔   𝑔,𝑊

Allowed substitution hints:   𝐵(𝑔)   𝑅(𝑔)   𝑆(𝑔)   𝐹(𝑔)   𝐻(𝑔)   𝐼(𝑔)   ≤ (𝑔)   𝑂(𝑔)   𝑉(𝑔)   𝑋(𝑔)

Proof of Theorem dihopelvalbN

Step	Hyp	Ref	Expression
1		dihval3.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		dihval3.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		dihval3.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
4		dihval3.i	. . . 4 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
5		eqid 2727	. . . 4 ⊢ ((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊)
6	1, 2, 3, 4, 5	dihvalb 40714	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = (((DIsoB‘𝐾)‘𝑊)‘𝑋))
7	6	eleq2d 2814	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼‘𝑋) ↔ ⟨𝐹, 𝑆⟩ ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑋)))
8		dihval3.t	. . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
9		dihval3.r	. . 3 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
10		dihval3.o	. . 3 ⊢ 𝑂 = (𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵))
11	1, 2, 3, 8, 9, 10, 5	dibopelval3 40625	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑋) ↔ ((𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑋) ∧ 𝑆 = 𝑂)))
12	7, 11	bitrd 278	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼‘𝑋) ↔ ((𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑋) ∧ 𝑆 = 𝑂)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ⟨cop 4636   class class class wbr 5150   ↦ cmpt 5233   I cid 5577   ↾ cres 5682  ‘cfv 6551  Basecbs 17185  lecple 17245  LHypclh 39461  LTrncltrn 39578  trLctrl 39635  DIsoBcdib 40615  DIsoHcdih 40705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-riota 7380  df-ov 7427  df-disoa 40506  df-dib 40616  df-dih 40706

This theorem is referenced by:  dihmeetlem1N  40767  dihglblem5apreN  40768  dihmeetlem4preN  40783