Theorem vrmdfval 18870

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 18864	. . 3 ⊢ varFMnd = (𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ ⟨“𝑗”⟩))
3		mpteq1 5234	. . 3 ⊢ (𝑖 = 𝐼 → (𝑗 ∈ 𝑖 ↦ ⟨“𝑗”⟩) = (𝑗 ∈ 𝐼 ↦ ⟨“𝑗”⟩))
4		elex 3500	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V)
5		mptexg 7242	. . 3 ⊢ (𝐼 ∈ 𝑉 → (𝑗 ∈ 𝐼 ↦ ⟨“𝑗”⟩) ∈ V)
6	2, 3, 4, 5	fvmptd3 7038	. 2 ⊢ (𝐼 ∈ 𝑉 → (varFMnd‘𝐼) = (𝑗 ∈ 𝐼 ↦ ⟨“𝑗”⟩))
7	1, 6	eqtrid 2788	1 ⊢ (𝐼 ∈ 𝑉 → 𝑈 = (𝑗 ∈ 𝐼 ↦ ⟨“𝑗”⟩))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  Vcvv 3479   ↦ cmpt 5224  ‘cfv 6560  ⟨“cs1 14634  var_FMndcvrmd 18862

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-vrmd 18864

This theorem is referenced by:  vrmdval  18871  vrmdf  18872  frgpup3lem  19796