Theorem vrmdfval 18410

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 = (varFMnd‘𝐼)

Assertion

Ref	Expression
vrmdfval	⊢ (𝐼 ∈ 𝑉 → 𝑈 = (𝑗 ∈ 𝐼 ↦ ⟨“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.

Step	Hyp	Ref	Expression
1		vrmdfval.u	. 2 ⊢ 𝑈 = (varFMnd‘𝐼)
2		df-vrmd 18404	. . 3 ⊢ varFMnd = (𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ ⟨“𝑗”⟩))
3		mpteq1 5163	. . 3 ⊢ (𝑖 = 𝐼 → (𝑗 ∈ 𝑖 ↦ ⟨“𝑗”⟩) = (𝑗 ∈ 𝐼 ↦ ⟨“𝑗”⟩))
4		elex 3440	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V)
5		mptexg 7079	. . 3 ⊢ (𝐼 ∈ 𝑉 → (𝑗 ∈ 𝐼 ↦ ⟨“𝑗”⟩) ∈ V)
6	2, 3, 4, 5	fvmptd3 6880	. 2 ⊢ (𝐼 ∈ 𝑉 → (varFMnd‘𝐼) = (𝑗 ∈ 𝐼 ↦ ⟨“𝑗”⟩))
7	1, 6	eqtrid 2790	1 ⊢ (𝐼 ∈ 𝑉 → 𝑈 = (𝑗 ∈ 𝐼 ↦ ⟨“𝑗”⟩))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ↦ cmpt 5153  ‘cfv 6418  ⟨“cs1 14228  var_FMndcvrmd 18402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-vrmd 18404

This theorem is referenced by:  vrmdval  18411  vrmdf  18412  frgpup3lem  19298