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Theorem lspfval 20817
Description: The span function for a left vector space (or a left module). (df-span 31066 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 3487 . . 3 (π‘Š ∈ 𝑋 β†’ π‘Š ∈ V)
3 fveq2 6884 . . . . . . 7 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = (Baseβ€˜π‘Š))
4 lspval.v . . . . . . 7 𝑉 = (Baseβ€˜π‘Š)
53, 4eqtr4di 2784 . . . . . 6 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = 𝑉)
65pweqd 4614 . . . . 5 (𝑀 = π‘Š β†’ 𝒫 (Baseβ€˜π‘€) = 𝒫 𝑉)
7 fveq2 6884 . . . . . . . 8 (𝑀 = π‘Š β†’ (LSubSpβ€˜π‘€) = (LSubSpβ€˜π‘Š))
8 lspval.s . . . . . . . 8 𝑆 = (LSubSpβ€˜π‘Š)
97, 8eqtr4di 2784 . . . . . . 7 (𝑀 = π‘Š β†’ (LSubSpβ€˜π‘€) = 𝑆)
109rabeqdv 3441 . . . . . 6 (𝑀 = π‘Š β†’ {𝑑 ∈ (LSubSpβ€˜π‘€) ∣ 𝑠 βŠ† 𝑑} = {𝑑 ∈ 𝑆 ∣ 𝑠 βŠ† 𝑑})
1110inteqd 4948 . . . . 5 (𝑀 = π‘Š β†’ ∩ {𝑑 ∈ (LSubSpβ€˜π‘€) ∣ 𝑠 βŠ† 𝑑} = ∩ {𝑑 ∈ 𝑆 ∣ 𝑠 βŠ† 𝑑})
126, 11mpteq12dv 5232 . . . 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 20816 . . . 4 LSpan = (𝑀 ∈ V ↦ (𝑠 ∈ 𝒫 (Baseβ€˜π‘€) ↦ ∩ {𝑑 ∈ (LSubSpβ€˜π‘€) ∣ 𝑠 βŠ† 𝑑}))
144fvexi 6898 . . . . . 6 𝑉 ∈ V
1514pwex 5371 . . . . 5 𝒫 𝑉 ∈ V
1615mptex 7219 . . . 4 (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ 𝑆 ∣ 𝑠 βŠ† 𝑑}) ∈ V
1712, 13, 16fvmpt 6991 . . 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 2778 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 3426  Vcvv 3468   βŠ† wss 3943  π’« cpw 4597  βˆ© cint 4943   ↦ cmpt 5224  β€˜cfv 6536  Basecbs 17150  LSubSpclss 20775  LSpanclspn 20815
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-pow 5356  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-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  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-lsp 20816
This theorem is referenced by:  lspf  20818  lspval  20819  00lsp  20825  mrclsp  20833  lsppropd  20863
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