Theorem dibelval3 38277

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 2821	. . . 4 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
7		dibval3.i	. . . 4 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)
8	1, 2, 3, 4, 5, 6, 7	dibval2 38274	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }))
9	8	eleq2d 2898	. 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 38163	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ↔ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑋)))
12	11	anbi1d 631	. . . . 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 645	. . . . . . . 8 ⊢ ((𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑠 ∈ { 0 })) ↔ (𝑠 ∈ { 0 } ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 𝑠⟩)))
14		velsn 4576	. . . . . . . . 9 ⊢ (𝑠 ∈ { 0 } ↔ 𝑠 = 0 )
15	14	anbi1i 625	. . . . . . . 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 277	. . . . . . 7 ⊢ ((𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑠 ∈ { 0 })) ↔ (𝑠 = 0 ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 𝑠⟩)))
17	16	exbii 1844	. . . . . 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 6678	. . . . . . . . 9 ⊢ 𝑇 ∈ V
19	18	mptex 6980	. . . . . . . 8 ⊢ (𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵)) ∈ V
20	5, 19	eqeltri 2909	. . . . . . 7 ⊢ 0 ∈ V
21		opeq2 4797	. . . . . . . . 9 ⊢ (𝑠 = 0 → ⟨𝑓, 𝑠⟩ = ⟨𝑓, 0 ⟩)
22	21	eqeq2d 2832	. . . . . . . 8 ⊢ (𝑠 = 0 → (𝑌 = ⟨𝑓, 𝑠⟩ ↔ 𝑌 = ⟨𝑓, 0 ⟩))
23	22	anbi2d 630	. . . . . . 7 ⊢ (𝑠 = 0 → ((𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 𝑠⟩) ↔ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 0 ⟩)))
24	20, 23	ceqsexv 3541	. . . . . 6 ⊢ (∃𝑠(𝑠 = 0 ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 𝑠⟩)) ↔ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 0 ⟩))
25	17, 24	bitri 277	. . . . 5 ⊢ (∃𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑠 ∈ { 0 })) ↔ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 0 ⟩))
26		anass 471	. . . . . 6 ⊢ (((𝑓 ∈ 𝑇 ∧ 𝑌 = ⟨𝑓, 0 ⟩) ∧ (𝑅‘𝑓) ≤ 𝑋) ↔ (𝑓 ∈ 𝑇 ∧ (𝑌 = ⟨𝑓, 0 ⟩ ∧ (𝑅‘𝑓) ≤ 𝑋)))
27		an32 644	. . . . . 6 ⊢ (((𝑓 ∈ 𝑇 ∧ 𝑌 = ⟨𝑓, 0 ⟩) ∧ (𝑅‘𝑓) ≤ 𝑋) ↔ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑋) ∧ 𝑌 = ⟨𝑓, 0 ⟩))
28	26, 27	bitr3i 279	. . . . 5 ⊢ ((𝑓 ∈ 𝑇 ∧ (𝑌 = ⟨𝑓, 0 ⟩ ∧ (𝑅‘𝑓) ≤ 𝑋)) ↔ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑋) ∧ 𝑌 = ⟨𝑓, 0 ⟩))
29	12, 25, 28	3bitr4g 316	. . . 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 1918	. . 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 5572	. . 3 ⊢ (𝑌 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }) ↔ ∃𝑓∃𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑠 ∈ { 0 })))
32		df-rex 3144	. . 3 ⊢ (∃𝑓 ∈ 𝑇 (𝑌 = ⟨𝑓, 0 ⟩ ∧ (𝑅‘𝑓) ≤ 𝑋) ↔ ∃𝑓(𝑓 ∈ 𝑇 ∧ (𝑌 = ⟨𝑓, 0 ⟩ ∧ (𝑅‘𝑓) ≤ 𝑋)))
33	30, 31, 32	3bitr4g 316	. 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 281	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 208   ∧ wa 398   = wceq 1533  ∃wex 1776   ∈ wcel 2110  ∃wrex 3139  Vcvv 3494  {csn 4560  ⟨cop 4566   class class class wbr 5058   ↦ cmpt 5138   I cid 5453   × cxp 5547   ↾ cres 5551  ‘cfv 6349  Basecbs 16477  lecple 16566  LHypclh 37114  LTrncltrn 37231  trLctrl 37288  DIsoAcdia 38158  DIsoBcdib 38268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-disoa 38159  df-dib 38269

This theorem is referenced by:  cdlemn11pre  38340  dihord2pre  38355