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Theorem vrmdfval 18021
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 18015 . . 3 varFMnd = (𝑖 ∈ V ↦ (𝑗𝑖 ↦ ⟨“𝑗”⟩))
3 mpteq1 5140 . . 3 (𝑖 = 𝐼 → (𝑗𝑖 ↦ ⟨“𝑗”⟩) = (𝑗𝐼 ↦ ⟨“𝑗”⟩))
4 elex 3498 . . 3 (𝐼𝑉𝐼 ∈ V)
5 mptexg 6975 . . 3 (𝐼𝑉 → (𝑗𝐼 ↦ ⟨“𝑗”⟩) ∈ V)
62, 3, 4, 5fvmptd3 6782 . 2 (𝐼𝑉 → (varFMnd𝐼) = (𝑗𝐼 ↦ ⟨“𝑗”⟩))
71, 6syl5eq 2871 1 (𝐼𝑉𝑈 = (𝑗𝐼 ↦ ⟨“𝑗”⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  Vcvv 3480  cmpt 5132  cfv 6343  ⟨“cs1 13949  varFMndcvrmd 18013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-vrmd 18015
This theorem is referenced by:  vrmdval  18022  vrmdf  18023  frgpup3lem  18903
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