Theorem dibelval3 42023

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 2760	. . . 4 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
7		dibval3.i	. . . 4 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)
8	1, 2, 3, 4, 5, 6, 7	dibval2 42020	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }))
9	8	eleq2d 2846	. 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 41909	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ↔ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑋)))
12	11	anbi1d 643	. . . . 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 660	. . . . . . . 8 ⊢ ((𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑠 ∈ { 0 })) ↔ (𝑠 ∈ { 0 } ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 𝑠⟩)))
14		velsn 4600	. . . . . . . . 9 ⊢ (𝑠 ∈ { 0 } ↔ 𝑠 = 0 )
15	14	anbi1i 636	. . . . . . . 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 278	. . . . . . 7 ⊢ ((𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑠 ∈ { 0 })) ↔ (𝑠 = 0 ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 𝑠⟩)))
17	16	exbii 1881	. . . . . 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 6893	. . . . . . . . 9 ⊢ 𝑇 ∈ V
19	18	mptex 7223	. . . . . . . 8 ⊢ (𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵)) ∈ V
20	5, 19	eqeltri 2856	. . . . . . 7 ⊢ 0 ∈ V
21		opeq2 4834	. . . . . . . . 9 ⊢ (𝑠 = 0 → ⟨𝑓, 𝑠⟩ = ⟨𝑓, 0 ⟩)
22	21	eqeq2d 2771	. . . . . . . 8 ⊢ (𝑠 = 0 → (𝑌 = ⟨𝑓, 𝑠⟩ ↔ 𝑌 = ⟨𝑓, 0 ⟩))
23	22	anbi2d 642	. . . . . . 7 ⊢ (𝑠 = 0 → ((𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 𝑠⟩) ↔ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 0 ⟩)))
24	20, 23	ceqsexv 3498	. . . . . 6 ⊢ (∃𝑠(𝑠 = 0 ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 𝑠⟩)) ↔ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 0 ⟩))
25	17, 24	bitri 278	. . . . 5 ⊢ (∃𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑠 ∈ { 0 })) ↔ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 0 ⟩))
26		anass 474	. . . . . 6 ⊢ (((𝑓 ∈ 𝑇 ∧ 𝑌 = ⟨𝑓, 0 ⟩) ∧ (𝑅‘𝑓) ≤ 𝑋) ↔ (𝑓 ∈ 𝑇 ∧ (𝑌 = ⟨𝑓, 0 ⟩ ∧ (𝑅‘𝑓) ≤ 𝑋)))
27		an32 659	. . . . . 6 ⊢ (((𝑓 ∈ 𝑇 ∧ 𝑌 = ⟨𝑓, 0 ⟩) ∧ (𝑅‘𝑓) ≤ 𝑋) ↔ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑋) ∧ 𝑌 = ⟨𝑓, 0 ⟩))
28	26, 27	bitr3i 280	. . . . 5 ⊢ ((𝑓 ∈ 𝑇 ∧ (𝑌 = ⟨𝑓, 0 ⟩ ∧ (𝑅‘𝑓) ≤ 𝑋)) ↔ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑋) ∧ 𝑌 = ⟨𝑓, 0 ⟩))
29	12, 25, 28	3bitr4g 317	. . . 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 1954	. . 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 5678	. . 3 ⊢ (𝑌 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }) ↔ ∃𝑓∃𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑠 ∈ { 0 })))
32		df-rex 3087	. . 3 ⊢ (∃𝑓 ∈ 𝑇 (𝑌 = ⟨𝑓, 0 ⟩ ∧ (𝑅‘𝑓) ≤ 𝑋) ↔ ∃𝑓(𝑓 ∈ 𝑇 ∧ (𝑌 = ⟨𝑓, 0 ⟩ ∧ (𝑅‘𝑓) ≤ 𝑋)))
33	30, 31, 32	3bitr4g 317	. 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 282	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑋) ↔ ∃𝑓 ∈ 𝑇 (𝑌 = ⟨𝑓, 0 ⟩ ∧ (𝑅‘𝑓) ≤ 𝑋)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃wrex 3086  Vcvv 3450  {csn 4584  ⟨cop 4590   class class class wbr 5103   ↦ cmpt 5186   I cid 5549   × cxp 5653   ↾ cres 5657  ‘cfv 6533  Basecbs 17304  lecple 17352  LHypclh 40860  LTrncltrn 40977  trLctrl 41034  DIsoAcdia 41904  DIsoBcdib 42014

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-disoa 41905  df-dib 42015

This theorem is used by:  cdlemn11pre  42086  dihord2pre  42101