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Theorem diaval 39524
Description: The partial isomorphism A for a lattice 𝐾. Definition of isomorphism map in [Crawley] p. 120 line 24. (Contributed by NM, 15-Oct-2013.)
Hypotheses
Ref Expression
diaval.b 𝐡 = (Baseβ€˜πΎ)
diaval.l ≀ = (leβ€˜πΎ)
diaval.h 𝐻 = (LHypβ€˜πΎ)
diaval.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
diaval.r 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
diaval.i 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
diaval (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (πΌβ€˜π‘‹) = {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ 𝑋})
Distinct variable groups:   𝑓,𝐾   𝑇,𝑓   𝑓,π‘Š   𝑓,𝑋
Allowed substitution hints:   𝐡(𝑓)   𝑅(𝑓)   𝐻(𝑓)   𝐼(𝑓)   ≀ (𝑓)   𝑉(𝑓)

Proof of Theorem diaval
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 diaval.b . . . . 5 𝐡 = (Baseβ€˜πΎ)
2 diaval.l . . . . 5 ≀ = (leβ€˜πΎ)
3 diaval.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
4 diaval.t . . . . 5 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
5 diaval.r . . . . 5 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
6 diaval.i . . . . 5 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
71, 2, 3, 4, 5, 6diafval 39523 . . . 4 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝐼 = (π‘₯ ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š} ↦ {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ π‘₯}))
87adantr 482 . . 3 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ 𝐼 = (π‘₯ ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š} ↦ {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ π‘₯}))
98fveq1d 6849 . 2 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (πΌβ€˜π‘‹) = ((π‘₯ ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š} ↦ {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ π‘₯})β€˜π‘‹))
10 simpr 486 . . . 4 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š))
11 breq1 5113 . . . . 5 (𝑦 = 𝑋 β†’ (𝑦 ≀ π‘Š ↔ 𝑋 ≀ π‘Š))
1211elrab 3650 . . . 4 (𝑋 ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š} ↔ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š))
1310, 12sylibr 233 . . 3 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ 𝑋 ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š})
14 breq2 5114 . . . . 5 (π‘₯ = 𝑋 β†’ ((π‘…β€˜π‘“) ≀ π‘₯ ↔ (π‘…β€˜π‘“) ≀ 𝑋))
1514rabbidv 3418 . . . 4 (π‘₯ = 𝑋 β†’ {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ π‘₯} = {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ 𝑋})
16 eqid 2737 . . . 4 (π‘₯ ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š} ↦ {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ π‘₯}) = (π‘₯ ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š} ↦ {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ π‘₯})
174fvexi 6861 . . . . 5 𝑇 ∈ V
1817rabex 5294 . . . 4 {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ 𝑋} ∈ V
1915, 16, 18fvmpt 6953 . . 3 (𝑋 ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š} β†’ ((π‘₯ ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š} ↦ {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ π‘₯})β€˜π‘‹) = {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ 𝑋})
2013, 19syl 17 . 2 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ ((π‘₯ ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š} ↦ {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ π‘₯})β€˜π‘‹) = {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ 𝑋})
219, 20eqtrd 2777 1 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (πΌβ€˜π‘‹) = {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ 𝑋})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {crab 3410   class class class wbr 5110   ↦ cmpt 5193  β€˜cfv 6501  Basecbs 17090  lecple 17147  LHypclh 38476  LTrncltrn 38593  trLctrl 38650  DIsoAcdia 39520
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 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389
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 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-disoa 39521
This theorem is referenced by:  diaelval  39525  diass  39534  diaord  39539  dia0  39544  dia1N  39545  diassdvaN  39552  dia1dim  39553  cdlemm10N  39610  dibval3N  39638  dihwN  39781
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