Theorem vrmdfval 19059

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 19053	. . 3 ⊢ varFMnd = (𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ ⟨“𝑗”⟩))
3		mpteq1 5193	. . 3 ⊢ (𝑖 = 𝐼 → (𝑗 ∈ 𝑖 ↦ ⟨“𝑗”⟩) = (𝑗 ∈ 𝐼 ↦ ⟨“𝑗”⟩))
4		elex 3471	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V)
5		mptexg 7229	. . 3 ⊢ (𝐼 ∈ 𝑉 → (𝑗 ∈ 𝐼 ↦ ⟨“𝑗”⟩) ∈ V)
6	2, 3, 4, 5	fvmptd3 7020	. 2 ⊢ (𝐼 ∈ 𝑉 → (varFMnd‘𝐼) = (𝑗 ∈ 𝐼 ↦ ⟨“𝑗”⟩))
7	1, 6	eqtrid 2807	1 ⊢ (𝐼 ∈ 𝑉 → 𝑈 = (𝑗 ∈ 𝐼 ↦ ⟨“𝑗”⟩))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  Vcvv 3450   ↦ cmpt 5185  ‘cfv 6541  ⟨“cs1 14747  var_FMndcvrmd 19051

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-vrmd 19053

This theorem is used by:  vrmdval  19060  vrmdf  19061  frgpup3lem  19998