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Theorem dibopelval2 39611
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 39610 . . 3 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (πΌβ€˜π‘‹) = ((π½β€˜π‘‹) Γ— { 0 }))
98eleq2d 2824 . 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 5670 . . 3 (⟨𝐹, π‘†βŸ© ∈ ((π½β€˜π‘‹) Γ— { 0 }) ↔ (𝐹 ∈ (π½β€˜π‘‹) ∧ 𝑆 ∈ { 0 }))
114fvexi 6857 . . . . . . 7 𝑇 ∈ V
1211mptex 7174 . . . . . 6 (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡)) ∈ V
135, 12eqeltri 2834 . . . . 5 0 ∈ V
1413elsn2 4626 . . . 4 (𝑆 ∈ { 0 } ↔ 𝑆 = 0 )
1514anbi2i 624 . . 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 397   = wceq 1542   ∈ wcel 2107  Vcvv 3446  {csn 4587  βŸ¨cop 4593   class class class wbr 5106   ↦ cmpt 5189   I cid 5531   Γ— cxp 5632   β†Ύ cres 5636  β€˜cfv 6497  Basecbs 17084  lecple 17141  LHypclh 38450  LTrncltrn 38567  DIsoAcdia 39494  DIsoBcdib 39604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-disoa 39495  df-dib 39605
This theorem is referenced by:  dibopelval3  39614  dibglbN  39632  diblsmopel  39637  dib2dim  39709  dih2dimbALTN  39711  dihord6apre  39722
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