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Theorem dibopelval2 40529
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 40528 . . 3 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (πΌβ€˜π‘‹) = ((π½β€˜π‘‹) Γ— { 0 }))
98eleq2d 2813 . 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 5705 . . 3 (⟨𝐹, π‘†βŸ© ∈ ((π½β€˜π‘‹) Γ— { 0 }) ↔ (𝐹 ∈ (π½β€˜π‘‹) ∧ 𝑆 ∈ { 0 }))
114fvexi 6899 . . . . . . 7 𝑇 ∈ V
1211mptex 7220 . . . . . 6 (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡)) ∈ V
135, 12eqeltri 2823 . . . . 5 0 ∈ V
1413elsn2 4662 . . . 4 (𝑆 ∈ { 0 } ↔ 𝑆 = 0 )
1514anbi2i 622 . . 3 ((𝐹 ∈ (π½β€˜π‘‹) ∧ 𝑆 ∈ { 0 }) ↔ (𝐹 ∈ (π½β€˜π‘‹) ∧ 𝑆 = 0 ))
1610, 15bitri 275 . 2 (⟨𝐹, π‘†βŸ© ∈ ((π½β€˜π‘‹) Γ— { 0 }) ↔ (𝐹 ∈ (π½β€˜π‘‹) ∧ 𝑆 = 0 ))
179, 16bitrdi 287 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 395   = wceq 1533   ∈ wcel 2098  Vcvv 3468  {csn 4623  βŸ¨cop 4629   class class class wbr 5141   ↦ cmpt 5224   I cid 5566   Γ— cxp 5667   β†Ύ cres 5671  β€˜cfv 6537  Basecbs 17153  lecple 17213  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-disoa 40413  df-dib 40523
This theorem is referenced by:  dibopelval3  40532  dibglbN  40550  diblsmopel  40555  dib2dim  40627  dih2dimbALTN  40629  dihord6apre  40640
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