Theorem dibelval3 41104

Description: Member of the partial isomorphism B. (Contributed by NM, 26-Feb-2014.)

Hypotheses

Ref	Expression
dibval3.b	⊢ 𝐵 = (Base‘𝐾)
dibval3.l	⊢ ≤ = (le‘𝐾)
dibval3.h	⊢ 𝐻 = (LHyp‘𝐾)
dibval3.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dibval3.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
dibval3.o	⊢ 0 = (𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵))
dibval3.i	⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)

Assertion

Ref	Expression
dibelval3	⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑋) ↔ ∃𝑓 ∈ 𝑇 (𝑌 = ⟨𝑓, 0 ⟩ ∧ (𝑅‘𝑓) ≤ 𝑋)))

Distinct variable groups:   𝑓,𝐾   𝑔,𝐾   𝑇,𝑓   𝑓,𝑊   𝑔,𝑊   𝑓,𝑋   ≤ ,𝑓   𝐵,𝑓   𝑓,𝐻   0 ,𝑓   𝑇,𝑔   𝑓,𝑉   𝑓,𝑌

Allowed substitution hints:   𝐵(𝑔)   𝑅(𝑓,𝑔)   𝐻(𝑔)   𝐼(𝑓,𝑔)   ≤ (𝑔)   𝑉(𝑔)   𝑋(𝑔)   𝑌(𝑔)   0 (𝑔)

Proof of Theorem dibelval3

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dibval3.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		dibval3.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		dibval3.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
4		dibval3.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
5		dibval3.o	. . . 4 ⊢ 0 = (𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵))
6		eqid 2740	. . . 4 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
7		dibval3.i	. . . 4 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)
8	1, 2, 3, 4, 5, 6, 7	dibval2 41101	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }))
9	8	eleq2d 2830	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑋) ↔ 𝑌 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 })))
10		dibval3.r	. . . . . . 7 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
11	1, 2, 3, 4, 10, 6	diaelval 40990	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ↔ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑋)))
12	11	anbi1d 630	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → ((𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 0 ⟩) ↔ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑋) ∧ 𝑌 = ⟨𝑓, 0 ⟩)))
13		an13 646	. . . . . . . 8 ⊢ ((𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑠 ∈ { 0 })) ↔ (𝑠 ∈ { 0 } ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 𝑠⟩)))
14		velsn 4664	. . . . . . . . 9 ⊢ (𝑠 ∈ { 0 } ↔ 𝑠 = 0 )
15	14	anbi1i 623	. . . . . . . 8 ⊢ ((𝑠 ∈ { 0 } ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 𝑠⟩)) ↔ (𝑠 = 0 ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 𝑠⟩)))
16	13, 15	bitri 275	. . . . . . 7 ⊢ ((𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑠 ∈ { 0 })) ↔ (𝑠 = 0 ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 𝑠⟩)))
17	16	exbii 1846	. . . . . 6 ⊢ (∃𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑠 ∈ { 0 })) ↔ ∃𝑠(𝑠 = 0 ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 𝑠⟩)))
18	4	fvexi 6934	. . . . . . . . 9 ⊢ 𝑇 ∈ V
19	18	mptex 7260	. . . . . . . 8 ⊢ (𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵)) ∈ V
20	5, 19	eqeltri 2840	. . . . . . 7 ⊢ 0 ∈ V
21		opeq2 4898	. . . . . . . . 9 ⊢ (𝑠 = 0 → ⟨𝑓, 𝑠⟩ = ⟨𝑓, 0 ⟩)
22	21	eqeq2d 2751	. . . . . . . 8 ⊢ (𝑠 = 0 → (𝑌 = ⟨𝑓, 𝑠⟩ ↔ 𝑌 = ⟨𝑓, 0 ⟩))
23	22	anbi2d 629	. . . . . . 7 ⊢ (𝑠 = 0 → ((𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 𝑠⟩) ↔ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 0 ⟩)))
24	20, 23	ceqsexv 3542	. . . . . 6 ⊢ (∃𝑠(𝑠 = 0 ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 𝑠⟩)) ↔ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 0 ⟩))
25	17, 24	bitri 275	. . . . 5 ⊢ (∃𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑠 ∈ { 0 })) ↔ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 0 ⟩))
26		anass 468	. . . . . 6 ⊢ (((𝑓 ∈ 𝑇 ∧ 𝑌 = ⟨𝑓, 0 ⟩) ∧ (𝑅‘𝑓) ≤ 𝑋) ↔ (𝑓 ∈ 𝑇 ∧ (𝑌 = ⟨𝑓, 0 ⟩ ∧ (𝑅‘𝑓) ≤ 𝑋)))
27		an32 645	. . . . . 6 ⊢ (((𝑓 ∈ 𝑇 ∧ 𝑌 = ⟨𝑓, 0 ⟩) ∧ (𝑅‘𝑓) ≤ 𝑋) ↔ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑋) ∧ 𝑌 = ⟨𝑓, 0 ⟩))
28	26, 27	bitr3i 277	. . . . 5 ⊢ ((𝑓 ∈ 𝑇 ∧ (𝑌 = ⟨𝑓, 0 ⟩ ∧ (𝑅‘𝑓) ≤ 𝑋)) ↔ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑋) ∧ 𝑌 = ⟨𝑓, 0 ⟩))
29	12, 25, 28	3bitr4g 314	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (∃𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑠 ∈ { 0 })) ↔ (𝑓 ∈ 𝑇 ∧ (𝑌 = ⟨𝑓, 0 ⟩ ∧ (𝑅‘𝑓) ≤ 𝑋))))
30	29	exbidv 1920	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (∃𝑓∃𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑠 ∈ { 0 })) ↔ ∃𝑓(𝑓 ∈ 𝑇 ∧ (𝑌 = ⟨𝑓, 0 ⟩ ∧ (𝑅‘𝑓) ≤ 𝑋))))
31		elxp 5723	. . 3 ⊢ (𝑌 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }) ↔ ∃𝑓∃𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑠 ∈ { 0 })))
32		df-rex 3077	. . 3 ⊢ (∃𝑓 ∈ 𝑇 (𝑌 = ⟨𝑓, 0 ⟩ ∧ (𝑅‘𝑓) ≤ 𝑋) ↔ ∃𝑓(𝑓 ∈ 𝑇 ∧ (𝑌 = ⟨𝑓, 0 ⟩ ∧ (𝑅‘𝑓) ≤ 𝑋)))
33	30, 31, 32	3bitr4g 314	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝑌 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }) ↔ ∃𝑓 ∈ 𝑇 (𝑌 = ⟨𝑓, 0 ⟩ ∧ (𝑅‘𝑓) ≤ 𝑋)))
34	9, 33	bitrd 279	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑋) ↔ ∃𝑓 ∈ 𝑇 (𝑌 = ⟨𝑓, 0 ⟩ ∧ (𝑅‘𝑓) ≤ 𝑋)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108  ∃wrex 3076  Vcvv 3488  {csn 4648  ⟨cop 4654   class class class wbr 5166   ↦ cmpt 5249   I cid 5592   × cxp 5698   ↾ cres 5702  ‘cfv 6573  Basecbs 17258  lecple 17318  LHypclh 39941  LTrncltrn 40058  trLctrl 40115  DIsoAcdia 40985  DIsoBcdib 41095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-disoa 40986  df-dib 41096

This theorem is referenced by:  cdlemn11pre  41167  dihord2pre  41182