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Theorem vrmdfval 18815
Description: The canonical injection from the generating set 𝐼 to the base set of the free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.)
Hypothesis
Ref Expression
vrmdfval.u 𝑈 = (varFMnd𝐼)
Assertion
Ref Expression
vrmdfval (𝐼𝑉𝑈 = (𝑗𝐼 ↦ ⟨“𝑗”⟩))
Distinct variable groups:   𝑗,𝐼   𝑗,𝑉
Allowed substitution hint:   𝑈(𝑗)

Proof of Theorem vrmdfval
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 vrmdfval.u . 2 𝑈 = (varFMnd𝐼)
2 df-vrmd 18809 . . 3 varFMnd = (𝑖 ∈ V ↦ (𝑗𝑖 ↦ ⟨“𝑗”⟩))
3 mpteq1 5245 . . 3 (𝑖 = 𝐼 → (𝑗𝑖 ↦ ⟨“𝑗”⟩) = (𝑗𝐼 ↦ ⟨“𝑗”⟩))
4 elex 3492 . . 3 (𝐼𝑉𝐼 ∈ V)
5 mptexg 7239 . . 3 (𝐼𝑉 → (𝑗𝐼 ↦ ⟨“𝑗”⟩) ∈ V)
62, 3, 4, 5fvmptd3 7033 . 2 (𝐼𝑉 → (varFMnd𝐼) = (𝑗𝐼 ↦ ⟨“𝑗”⟩))
71, 6eqtrid 2780 1 (𝐼𝑉𝑈 = (𝑗𝐼 ↦ ⟨“𝑗”⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3473  cmpt 5235  cfv 6553  ⟨“cs1 14585  varFMndcvrmd 18807
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-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-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  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-vrmd 18809
This theorem is referenced by:  vrmdval  18816  vrmdf  18817  frgpup3lem  19739
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