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Theorem lspfval 20864
Description: The span function for a left vector space (or a left module). (df-span 31139 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
Hypotheses
Ref Expression
lspval.v 𝑉 = (Baseβ€˜π‘Š)
lspval.s 𝑆 = (LSubSpβ€˜π‘Š)
lspval.n 𝑁 = (LSpanβ€˜π‘Š)
Assertion
Ref Expression
lspfval (π‘Š ∈ 𝑋 β†’ 𝑁 = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ 𝑆 ∣ 𝑠 βŠ† 𝑑}))
Distinct variable groups:   𝑑,𝑠,𝑆   𝑉,𝑠,𝑑   π‘Š,𝑠
Allowed substitution hints:   𝑁(𝑑,𝑠)   π‘Š(𝑑)   𝑋(𝑑,𝑠)

Proof of Theorem lspfval
Dummy variable 𝑀 is distinct from all other variables.
StepHypRef Expression
1 lspval.n . 2 𝑁 = (LSpanβ€˜π‘Š)
2 elex 3492 . . 3 (π‘Š ∈ 𝑋 β†’ π‘Š ∈ V)
3 fveq2 6902 . . . . . . 7 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = (Baseβ€˜π‘Š))
4 lspval.v . . . . . . 7 𝑉 = (Baseβ€˜π‘Š)
53, 4eqtr4di 2786 . . . . . 6 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = 𝑉)
65pweqd 4623 . . . . 5 (𝑀 = π‘Š β†’ 𝒫 (Baseβ€˜π‘€) = 𝒫 𝑉)
7 fveq2 6902 . . . . . . . 8 (𝑀 = π‘Š β†’ (LSubSpβ€˜π‘€) = (LSubSpβ€˜π‘Š))
8 lspval.s . . . . . . . 8 𝑆 = (LSubSpβ€˜π‘Š)
97, 8eqtr4di 2786 . . . . . . 7 (𝑀 = π‘Š β†’ (LSubSpβ€˜π‘€) = 𝑆)
109rabeqdv 3446 . . . . . 6 (𝑀 = π‘Š β†’ {𝑑 ∈ (LSubSpβ€˜π‘€) ∣ 𝑠 βŠ† 𝑑} = {𝑑 ∈ 𝑆 ∣ 𝑠 βŠ† 𝑑})
1110inteqd 4958 . . . . 5 (𝑀 = π‘Š β†’ ∩ {𝑑 ∈ (LSubSpβ€˜π‘€) ∣ 𝑠 βŠ† 𝑑} = ∩ {𝑑 ∈ 𝑆 ∣ 𝑠 βŠ† 𝑑})
126, 11mpteq12dv 5243 . . . 4 (𝑀 = π‘Š β†’ (𝑠 ∈ 𝒫 (Baseβ€˜π‘€) ↦ ∩ {𝑑 ∈ (LSubSpβ€˜π‘€) ∣ 𝑠 βŠ† 𝑑}) = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ 𝑆 ∣ 𝑠 βŠ† 𝑑}))
13 df-lsp 20863 . . . 4 LSpan = (𝑀 ∈ V ↦ (𝑠 ∈ 𝒫 (Baseβ€˜π‘€) ↦ ∩ {𝑑 ∈ (LSubSpβ€˜π‘€) ∣ 𝑠 βŠ† 𝑑}))
144fvexi 6916 . . . . . 6 𝑉 ∈ V
1514pwex 5384 . . . . 5 𝒫 𝑉 ∈ V
1615mptex 7241 . . . 4 (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ 𝑆 ∣ 𝑠 βŠ† 𝑑}) ∈ V
1712, 13, 16fvmpt 7010 . . 3 (π‘Š ∈ V β†’ (LSpanβ€˜π‘Š) = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ 𝑆 ∣ 𝑠 βŠ† 𝑑}))
182, 17syl 17 . 2 (π‘Š ∈ 𝑋 β†’ (LSpanβ€˜π‘Š) = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ 𝑆 ∣ 𝑠 βŠ† 𝑑}))
191, 18eqtrid 2780 1 (π‘Š ∈ 𝑋 β†’ 𝑁 = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ 𝑆 ∣ 𝑠 βŠ† 𝑑}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  {crab 3430  Vcvv 3473   βŠ† wss 3949  π’« cpw 4606  βˆ© cint 4953   ↦ cmpt 5235  β€˜cfv 6553  Basecbs 17187  LSubSpclss 20822  LSpanclspn 20862
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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-lsp 20863
This theorem is referenced by:  lspf  20865  lspval  20866  00lsp  20872  mrclsp  20880  lsppropd  20910
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