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Theorem dibopelval3 39614
Description: Member of the partial isomorphism B. (Contributed by NM, 3-Mar-2014.)
Hypotheses
Ref Expression
dibval3.b 𝐡 = (Baseβ€˜πΎ)
dibval3.l ≀ = (leβ€˜πΎ)
dibval3.h 𝐻 = (LHypβ€˜πΎ)
dibval3.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
dibval3.r 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
dibval3.o 0 = (𝑔 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))
dibval3.i 𝐼 = ((DIsoBβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
dibopelval3 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (⟨𝐹, π‘†βŸ© ∈ (πΌβ€˜π‘‹) ↔ ((𝐹 ∈ 𝑇 ∧ (π‘…β€˜πΉ) ≀ 𝑋) ∧ 𝑆 = 0 )))
Distinct variable groups:   𝑔,𝐾   𝑔,π‘Š   𝑇,𝑔
Allowed substitution hints:   𝐡(𝑔)   𝑅(𝑔)   𝑆(𝑔)   𝐹(𝑔)   𝐻(𝑔)   𝐼(𝑔)   ≀ (𝑔)   𝑉(𝑔)   𝑋(𝑔)   0 (𝑔)

Proof of Theorem dibopelval3
StepHypRef Expression
1 dibval3.b . . 3 𝐡 = (Baseβ€˜πΎ)
2 dibval3.l . . 3 ≀ = (leβ€˜πΎ)
3 dibval3.h . . 3 𝐻 = (LHypβ€˜πΎ)
4 dibval3.t . . 3 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
5 dibval3.o . . 3 0 = (𝑔 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))
6 eqid 2737 . . 3 ((DIsoAβ€˜πΎ)β€˜π‘Š) = ((DIsoAβ€˜πΎ)β€˜π‘Š)
7 dibval3.i . . 3 𝐼 = ((DIsoBβ€˜πΎ)β€˜π‘Š)
81, 2, 3, 4, 5, 6, 7dibopelval2 39611 . 2 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (⟨𝐹, π‘†βŸ© ∈ (πΌβ€˜π‘‹) ↔ (𝐹 ∈ (((DIsoAβ€˜πΎ)β€˜π‘Š)β€˜π‘‹) ∧ 𝑆 = 0 )))
9 dibval3.r . . . 4 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
101, 2, 3, 4, 9, 6diaelval 39499 . . 3 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (𝐹 ∈ (((DIsoAβ€˜πΎ)β€˜π‘Š)β€˜π‘‹) ↔ (𝐹 ∈ 𝑇 ∧ (π‘…β€˜πΉ) ≀ 𝑋)))
1110anbi1d 631 . 2 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ ((𝐹 ∈ (((DIsoAβ€˜πΎ)β€˜π‘Š)β€˜π‘‹) ∧ 𝑆 = 0 ) ↔ ((𝐹 ∈ 𝑇 ∧ (π‘…β€˜πΉ) ≀ 𝑋) ∧ 𝑆 = 0 )))
128, 11bitrd 279 1 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (⟨𝐹, π‘†βŸ© ∈ (πΌβ€˜π‘‹) ↔ ((𝐹 ∈ 𝑇 ∧ (π‘…β€˜πΉ) ≀ 𝑋) ∧ 𝑆 = 0 )))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βŸ¨cop 4593   class class class wbr 5106   ↦ cmpt 5189   I cid 5531   β†Ύ cres 5636  β€˜cfv 6497  Basecbs 17084  lecple 17141  LHypclh 38450  LTrncltrn 38567  trLctrl 38624  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:  dihord2cN  39687  dihord11b  39688  dihopelvalbN  39704  dihopelvalcpre  39714  dihjatcclem4  39887
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