Theorem dibopelvalN 40527

Description: Member of the partial isomorphism B. (Contributed by NM, 18-Jan-2014.) (Revised by Mario Carneiro, 6-May-2015.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dibval.b	⊢ 𝐵 = (Base‘𝐾)
dibval.h	⊢ 𝐻 = (LHyp‘𝐾)
dibval.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dibval.o	⊢ 0 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))
dibval.j	⊢ 𝐽 = ((DIsoA‘𝐾)‘𝑊)
dibval.i	⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)

Assertion

Ref	Expression
dibopelvalN	⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐽) → (⟨𝐹, 𝑆⟩ ∈ (𝐼‘𝑋) ↔ (𝐹 ∈ (𝐽‘𝑋) ∧ 𝑆 = 0 )))

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

Allowed substitution hints:   𝐵(𝑓)   𝑆(𝑓)   𝐹(𝑓)   𝐻(𝑓)   𝐼(𝑓)   𝐽(𝑓)   𝑉(𝑓)   𝑋(𝑓)   0 (𝑓)

Proof of Theorem dibopelvalN

Step	Hyp	Ref	Expression
1		dibval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		dibval.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
3		dibval.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
4		dibval.o	. . . 4 ⊢ 0 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))
5		dibval.j	. . . 4 ⊢ 𝐽 = ((DIsoA‘𝐾)‘𝑊)
6		dibval.i	. . . 4 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)
7	1, 2, 3, 4, 5, 6	dibval 40526	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐽) → (𝐼‘𝑋) = ((𝐽‘𝑋) × { 0 }))
8	7	eleq2d 2813	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐽) → (⟨𝐹, 𝑆⟩ ∈ (𝐼‘𝑋) ↔ ⟨𝐹, 𝑆⟩ ∈ ((𝐽‘𝑋) × { 0 })))
9		opelxp 5705	. . 3 ⊢ (⟨𝐹, 𝑆⟩ ∈ ((𝐽‘𝑋) × { 0 }) ↔ (𝐹 ∈ (𝐽‘𝑋) ∧ 𝑆 ∈ { 0 }))
10	3	fvexi 6899	. . . . . . 7 ⊢ 𝑇 ∈ V
11	10	mptex 7220	. . . . . 6 ⊢ (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) ∈ V
12	4, 11	eqeltri 2823	. . . . 5 ⊢ 0 ∈ V
13	12	elsn2 4662	. . . 4 ⊢ (𝑆 ∈ { 0 } ↔ 𝑆 = 0 )
14	13	anbi2i 622	. . 3 ⊢ ((𝐹 ∈ (𝐽‘𝑋) ∧ 𝑆 ∈ { 0 }) ↔ (𝐹 ∈ (𝐽‘𝑋) ∧ 𝑆 = 0 ))
15	9, 14	bitri 275	. 2 ⊢ (⟨𝐹, 𝑆⟩ ∈ ((𝐽‘𝑋) × { 0 }) ↔ (𝐹 ∈ (𝐽‘𝑋) ∧ 𝑆 = 0 ))
16	8, 15	bitrdi 287	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐽) → (⟨𝐹, 𝑆⟩ ∈ (𝐼‘𝑋) ↔ (𝐹 ∈ (𝐽‘𝑋) ∧ 𝑆 = 0 )))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3468  {csn 4623  ⟨cop 4629   ↦ cmpt 5224   I cid 5566   × cxp 5667  dom cdm 5669   ↾ cres 5671  ‘cfv 6537  Basecbs 17153  LHypclh 39368  LTrncltrn 39485  DIsoAcdia 40412  DIsoBcdib 40522

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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-dib 40523

This theorem is referenced by: (None)