Theorem dihopelvalbN 37308

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 2825	. . . 4 ⊢ ((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊)
6	1, 2, 3, 4, 5	dihvalb 37307	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = (((DIsoB‘𝐾)‘𝑊)‘𝑋))
7	6	eleq2d 2892	. 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 37218	. 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 271	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 198   ∧ wa 386   = wceq 1656   ∈ wcel 2164  ⟨cop 4405   class class class wbr 4875   ↦ cmpt 4954   I cid 5251   ↾ cres 5348  ‘cfv 6127  Basecbs 16229  lecple 16319  LHypclh 36054  LTrncltrn 36171  trLctrl 36228  DIsoBcdib 37208  DIsoHcdih 37298

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-disoa 37099  df-dib 37209  df-dih 37299

This theorem is referenced by:  dihmeetlem1N  37360  dihglblem5apreN  37361  dihmeetlem4preN  37376