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Theorem diafval 39523
Description: The partial isomorphism A for a lattice 𝐾. (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
diafval ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝐼 = (π‘₯ ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š} ↦ {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ π‘₯}))
Distinct variable groups:   π‘₯,𝑦, ≀   π‘₯,𝐡,𝑦   π‘₯,𝑓,𝑦,𝐾   π‘₯,𝑅   𝑇,𝑓,π‘₯   𝑓,π‘Š,π‘₯,𝑦
Allowed substitution hints:   𝐡(𝑓)   𝑅(𝑦,𝑓)   𝑇(𝑦)   𝐻(π‘₯,𝑦,𝑓)   𝐼(π‘₯,𝑦,𝑓)   ≀ (𝑓)   𝑉(π‘₯,𝑦,𝑓)

Proof of Theorem diafval
Dummy variable 𝑀 is distinct from all other variables.
StepHypRef Expression
1 diaval.i . . 3 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
2 diaval.b . . . . 5 𝐡 = (Baseβ€˜πΎ)
3 diaval.l . . . . 5 ≀ = (leβ€˜πΎ)
4 diaval.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
52, 3, 4diaffval 39522 . . . 4 (𝐾 ∈ 𝑉 β†’ (DIsoAβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ 𝑀} ↦ {𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ∣ (((trLβ€˜πΎ)β€˜π‘€)β€˜π‘“) ≀ π‘₯})))
65fveq1d 6849 . . 3 (𝐾 ∈ 𝑉 β†’ ((DIsoAβ€˜πΎ)β€˜π‘Š) = ((𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ 𝑀} ↦ {𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ∣ (((trLβ€˜πΎ)β€˜π‘€)β€˜π‘“) ≀ π‘₯}))β€˜π‘Š))
71, 6eqtrid 2789 . 2 (𝐾 ∈ 𝑉 β†’ 𝐼 = ((𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ 𝑀} ↦ {𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ∣ (((trLβ€˜πΎ)β€˜π‘€)β€˜π‘“) ≀ π‘₯}))β€˜π‘Š))
8 breq2 5114 . . . . 5 (𝑀 = π‘Š β†’ (𝑦 ≀ 𝑀 ↔ 𝑦 ≀ π‘Š))
98rabbidv 3418 . . . 4 (𝑀 = π‘Š β†’ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ 𝑀} = {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š})
10 fveq2 6847 . . . . . 6 (𝑀 = π‘Š β†’ ((LTrnβ€˜πΎ)β€˜π‘€) = ((LTrnβ€˜πΎ)β€˜π‘Š))
11 diaval.t . . . . . 6 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
1210, 11eqtr4di 2795 . . . . 5 (𝑀 = π‘Š β†’ ((LTrnβ€˜πΎ)β€˜π‘€) = 𝑇)
13 fveq2 6847 . . . . . . . 8 (𝑀 = π‘Š β†’ ((trLβ€˜πΎ)β€˜π‘€) = ((trLβ€˜πΎ)β€˜π‘Š))
14 diaval.r . . . . . . . 8 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
1513, 14eqtr4di 2795 . . . . . . 7 (𝑀 = π‘Š β†’ ((trLβ€˜πΎ)β€˜π‘€) = 𝑅)
1615fveq1d 6849 . . . . . 6 (𝑀 = π‘Š β†’ (((trLβ€˜πΎ)β€˜π‘€)β€˜π‘“) = (π‘…β€˜π‘“))
1716breq1d 5120 . . . . 5 (𝑀 = π‘Š β†’ ((((trLβ€˜πΎ)β€˜π‘€)β€˜π‘“) ≀ π‘₯ ↔ (π‘…β€˜π‘“) ≀ π‘₯))
1812, 17rabeqbidv 3427 . . . 4 (𝑀 = π‘Š β†’ {𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ∣ (((trLβ€˜πΎ)β€˜π‘€)β€˜π‘“) ≀ π‘₯} = {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ π‘₯})
199, 18mpteq12dv 5201 . . 3 (𝑀 = π‘Š β†’ (π‘₯ ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ 𝑀} ↦ {𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ∣ (((trLβ€˜πΎ)β€˜π‘€)β€˜π‘“) ≀ π‘₯}) = (π‘₯ ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š} ↦ {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ π‘₯}))
20 eqid 2737 . . 3 (𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ 𝑀} ↦ {𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ∣ (((trLβ€˜πΎ)β€˜π‘€)β€˜π‘“) ≀ π‘₯})) = (𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ 𝑀} ↦ {𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ∣ (((trLβ€˜πΎ)β€˜π‘€)β€˜π‘“) ≀ π‘₯}))
212fvexi 6861 . . . 4 𝐡 ∈ V
2221mptrabex 7180 . . 3 (π‘₯ ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š} ↦ {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ π‘₯}) ∈ V
2319, 20, 22fvmpt 6953 . 2 (π‘Š ∈ 𝐻 β†’ ((𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ 𝑀} ↦ {𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ∣ (((trLβ€˜πΎ)β€˜π‘€)β€˜π‘“) ≀ π‘₯}))β€˜π‘Š) = (π‘₯ ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š} ↦ {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ π‘₯}))
247, 23sylan9eq 2797 1 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝐼 = (π‘₯ ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š} ↦ {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ π‘₯}))
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:  diaval  39524  diafn  39526
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