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Theorem diafval 40414
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 40413 . . . 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 6886 . . 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 2778 . 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 5145 . . . . 5 (𝑀 = π‘Š β†’ (𝑦 ≀ 𝑀 ↔ 𝑦 ≀ π‘Š))
98rabbidv 3434 . . . 4 (𝑀 = π‘Š β†’ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ 𝑀} = {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š})
10 fveq2 6884 . . . . . 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 2784 . . . . 5 (𝑀 = π‘Š β†’ ((LTrnβ€˜πΎ)β€˜π‘€) = 𝑇)
13 fveq2 6884 . . . . . . . 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 2784 . . . . . . 7 (𝑀 = π‘Š β†’ ((trLβ€˜πΎ)β€˜π‘€) = 𝑅)
1615fveq1d 6886 . . . . . 6 (𝑀 = π‘Š β†’ (((trLβ€˜πΎ)β€˜π‘€)β€˜π‘“) = (π‘…β€˜π‘“))
1716breq1d 5151 . . . . 5 (𝑀 = π‘Š β†’ ((((trLβ€˜πΎ)β€˜π‘€)β€˜π‘“) ≀ π‘₯ ↔ (π‘…β€˜π‘“) ≀ π‘₯))
1812, 17rabeqbidv 3443 . . . 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 5232 . . 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 2726 . . 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 6898 . . . 4 𝐡 ∈ V
2221mptrabex 7221 . . 3 (π‘₯ ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š} ↦ {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ π‘₯}) ∈ V
2319, 20, 22fvmpt 6991 . 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 2786 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 395   = wceq 1533   ∈ wcel 2098  {crab 3426   class class class wbr 5141   ↦ cmpt 5224  β€˜cfv 6536  Basecbs 17150  lecple 17210  LHypclh 39367  LTrncltrn 39484  trLctrl 39541  DIsoAcdia 40411
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-pr 5420
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-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 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-disoa 40412
This theorem is referenced by:  diaval  40415  diafn  40417
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