Theorem frgpval 17994

Description: Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-2015.)

Hypotheses

Ref	Expression
frgpval.m	⊢ 𝐺 = (freeGrp‘𝐼)
frgpval.b	⊢ 𝑀 = (freeMnd‘(𝐼 × 2𝑜))
frgpval.r	⊢ ∼ = ( ~FG ‘𝐼)

Assertion

Ref	Expression
frgpval	⊢ (𝐼 ∈ 𝑉 → 𝐺 = (𝑀 /s ∼ ))

Proof of Theorem frgpval

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frgpval.m	. 2 ⊢ 𝐺 = (freeGrp‘𝐼)
2		elex 3185	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V)
3		xpeq1 5052	. . . . . . 7 ⊢ (𝑖 = 𝐼 → (𝑖 × 2𝑜) = (𝐼 × 2𝑜))
4	3	fveq2d 6107	. . . . . 6 ⊢ (𝑖 = 𝐼 → (freeMnd‘(𝑖 × 2𝑜)) = (freeMnd‘(𝐼 × 2𝑜)))
5		frgpval.b	. . . . . 6 ⊢ 𝑀 = (freeMnd‘(𝐼 × 2𝑜))
6	4, 5	syl6eqr 2662	. . . . 5 ⊢ (𝑖 = 𝐼 → (freeMnd‘(𝑖 × 2𝑜)) = 𝑀)
7		fveq2 6103	. . . . . 6 ⊢ (𝑖 = 𝐼 → ( ~FG ‘𝑖) = ( ~FG ‘𝐼))
8		frgpval.r	. . . . . 6 ⊢ ∼ = ( ~FG ‘𝐼)
9	7, 8	syl6eqr 2662	. . . . 5 ⊢ (𝑖 = 𝐼 → ( ~FG ‘𝑖) = ∼ )
10	6, 9	oveq12d 6567	. . . 4 ⊢ (𝑖 = 𝐼 → ((freeMnd‘(𝑖 × 2𝑜)) /s ( ~FG ‘𝑖)) = (𝑀 /s ∼ ))
11		df-frgp 17946	. . . 4 ⊢ freeGrp = (𝑖 ∈ V ↦ ((freeMnd‘(𝑖 × 2𝑜)) /s ( ~FG ‘𝑖)))
12		ovex 6577	. . . 4 ⊢ (𝑀 /s ∼ ) ∈ V
13	10, 11, 12	fvmpt 6191	. . 3 ⊢ (𝐼 ∈ V → (freeGrp‘𝐼) = (𝑀 /s ∼ ))
14	2, 13	syl 17	. 2 ⊢ (𝐼 ∈ 𝑉 → (freeGrp‘𝐼) = (𝑀 /s ∼ ))
15	1, 14	syl5eq 2656	1 ⊢ (𝐼 ∈ 𝑉 → 𝐺 = (𝑀 /s ∼ ))

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173   × cxp 5036  ‘cfv 5804  (class class class)co 6549  2_𝑜c2o 7441   /_s cqus 15988  freeMndcfrmd 17207   ~_FG cefg 17942  freeGrpcfrgp 17943

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-frgp 17946

This theorem is referenced by:  frgp0  17996  frgpeccl  17997  frgpadd  17999  frgpupf  18009  frgpup1  18011  frgpup3lem  18013  frgpnabllem2  18100