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Theorem vrmdfval 17780
 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 17774 . . 3 varFMnd = (𝑖 ∈ V ↦ (𝑗𝑖 ↦ ⟨“𝑗”⟩))
3 mpteq1 4972 . . 3 (𝑖 = 𝐼 → (𝑗𝑖 ↦ ⟨“𝑗”⟩) = (𝑗𝐼 ↦ ⟨“𝑗”⟩))
4 elex 3414 . . 3 (𝐼𝑉𝐼 ∈ V)
5 mptexg 6756 . . 3 (𝐼𝑉 → (𝑗𝐼 ↦ ⟨“𝑗”⟩) ∈ V)
62, 3, 4, 5fvmptd3 6564 . 2 (𝐼𝑉 → (varFMnd𝐼) = (𝑗𝐼 ↦ ⟨“𝑗”⟩))
71, 6syl5eq 2826 1 (𝐼𝑉𝑈 = (𝑗𝐼 ↦ ⟨“𝑗”⟩))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1601   ∈ wcel 2107  Vcvv 3398   ↦ cmpt 4965  ‘cfv 6135  ⟨“cs1 13685  varFMndcvrmd 17772 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pr 5138 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-vrmd 17774 This theorem is referenced by:  vrmdval  17781  vrmdf  17782  frgpup3lem  18576
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