Theorem vrmdfval 18778

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 18772	. . 3 ⊢ varFMnd = (𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ ⟨“𝑗”⟩))
3		mpteq1 5234	. . 3 ⊢ (𝑖 = 𝐼 → (𝑗 ∈ 𝑖 ↦ ⟨“𝑗”⟩) = (𝑗 ∈ 𝐼 ↦ ⟨“𝑗”⟩))
4		elex 3487	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V)
5		mptexg 7217	. . 3 ⊢ (𝐼 ∈ 𝑉 → (𝑗 ∈ 𝐼 ↦ ⟨“𝑗”⟩) ∈ V)
6	2, 3, 4, 5	fvmptd3 7014	. 2 ⊢ (𝐼 ∈ 𝑉 → (varFMnd‘𝐼) = (𝑗 ∈ 𝐼 ↦ ⟨“𝑗”⟩))
7	1, 6	eqtrid 2778	1 ⊢ (𝐼 ∈ 𝑉 → 𝑈 = (𝑗 ∈ 𝐼 ↦ ⟨“𝑗”⟩))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  Vcvv 3468   ↦ cmpt 5224  ‘cfv 6536  ⟨“cs1 14548  var_FMndcvrmd 18770

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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-vrmd 18772

This theorem is referenced by:  vrmdval  18779  vrmdf  18780  frgpup3lem  19694