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Theorem dibopelval2 40673
Description: Member of the partial isomorphism B. (Contributed by NM, 3-Mar-2014.) (Revised by Mario Carneiro, 6-May-2015.)
Hypotheses
Ref Expression
dibval2.b 𝐡 = (Baseβ€˜πΎ)
dibval2.l ≀ = (leβ€˜πΎ)
dibval2.h 𝐻 = (LHypβ€˜πΎ)
dibval2.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
dibval2.o 0 = (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))
dibval2.j 𝐽 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
dibval2.i 𝐼 = ((DIsoBβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
dibopelval2 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (⟨𝐹, π‘†βŸ© ∈ (πΌβ€˜π‘‹) ↔ (𝐹 ∈ (π½β€˜π‘‹) ∧ 𝑆 = 0 )))
Distinct variable groups:   𝑓,𝐾   𝑓,π‘Š   𝑇,𝑓
Allowed substitution hints:   𝐡(𝑓)   𝑆(𝑓)   𝐹(𝑓)   𝐻(𝑓)   𝐼(𝑓)   𝐽(𝑓)   ≀ (𝑓)   𝑉(𝑓)   𝑋(𝑓)   0 (𝑓)

Proof of Theorem dibopelval2
StepHypRef Expression
1 dibval2.b . . . 4 𝐡 = (Baseβ€˜πΎ)
2 dibval2.l . . . 4 ≀ = (leβ€˜πΎ)
3 dibval2.h . . . 4 𝐻 = (LHypβ€˜πΎ)
4 dibval2.t . . . 4 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
5 dibval2.o . . . 4 0 = (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))
6 dibval2.j . . . 4 𝐽 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
7 dibval2.i . . . 4 𝐼 = ((DIsoBβ€˜πΎ)β€˜π‘Š)
81, 2, 3, 4, 5, 6, 7dibval2 40672 . . 3 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (πΌβ€˜π‘‹) = ((π½β€˜π‘‹) Γ— { 0 }))
98eleq2d 2811 . 2 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (⟨𝐹, π‘†βŸ© ∈ (πΌβ€˜π‘‹) ↔ ⟨𝐹, π‘†βŸ© ∈ ((π½β€˜π‘‹) Γ— { 0 })))
10 opelxp 5708 . . 3 (⟨𝐹, π‘†βŸ© ∈ ((π½β€˜π‘‹) Γ— { 0 }) ↔ (𝐹 ∈ (π½β€˜π‘‹) ∧ 𝑆 ∈ { 0 }))
114fvexi 6905 . . . . . . 7 𝑇 ∈ V
1211mptex 7230 . . . . . 6 (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡)) ∈ V
135, 12eqeltri 2821 . . . . 5 0 ∈ V
1413elsn2 4663 . . . 4 (𝑆 ∈ { 0 } ↔ 𝑆 = 0 )
1514anbi2i 621 . . 3 ((𝐹 ∈ (π½β€˜π‘‹) ∧ 𝑆 ∈ { 0 }) ↔ (𝐹 ∈ (π½β€˜π‘‹) ∧ 𝑆 = 0 ))
1610, 15bitri 274 . 2 (⟨𝐹, π‘†βŸ© ∈ ((π½β€˜π‘‹) Γ— { 0 }) ↔ (𝐹 ∈ (π½β€˜π‘‹) ∧ 𝑆 = 0 ))
179, 16bitrdi 286 1 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (⟨𝐹, π‘†βŸ© ∈ (πΌβ€˜π‘‹) ↔ (𝐹 ∈ (π½β€˜π‘‹) ∧ 𝑆 = 0 )))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  Vcvv 3463  {csn 4624  βŸ¨cop 4630   class class class wbr 5143   ↦ cmpt 5226   I cid 5569   Γ— cxp 5670   β†Ύ cres 5674  β€˜cfv 6542  Basecbs 17177  lecple 17237  LHypclh 39512  LTrncltrn 39629  DIsoAcdia 40556  DIsoBcdib 40666
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 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-disoa 40557  df-dib 40667
This theorem is referenced by:  dibopelval3  40676  dibglbN  40694  diblsmopel  40699  dib2dim  40771  dih2dimbALTN  40773  dihord6apre  40784
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