Theorem frgpval 18171

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 3212	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V)
3		xpeq1 5128	. . . . . . 7 ⊢ (𝑖 = 𝐼 → (𝑖 × 2𝑜) = (𝐼 × 2𝑜))
4	3	fveq2d 6195	. . . . . 6 ⊢ (𝑖 = 𝐼 → (freeMnd‘(𝑖 × 2𝑜)) = (freeMnd‘(𝐼 × 2𝑜)))
5		frgpval.b	. . . . . 6 ⊢ 𝑀 = (freeMnd‘(𝐼 × 2𝑜))
6	4, 5	syl6eqr 2674	. . . . 5 ⊢ (𝑖 = 𝐼 → (freeMnd‘(𝑖 × 2𝑜)) = 𝑀)
7		fveq2 6191	. . . . . 6 ⊢ (𝑖 = 𝐼 → ( ~FG ‘𝑖) = ( ~FG ‘𝐼))
8		frgpval.r	. . . . . 6 ⊢ ∼ = ( ~FG ‘𝐼)
9	7, 8	syl6eqr 2674	. . . . 5 ⊢ (𝑖 = 𝐼 → ( ~FG ‘𝑖) = ∼ )
10	6, 9	oveq12d 6668	. . . 4 ⊢ (𝑖 = 𝐼 → ((freeMnd‘(𝑖 × 2𝑜)) /s ( ~FG ‘𝑖)) = (𝑀 /s ∼ ))
11		df-frgp 18123	. . . 4 ⊢ freeGrp = (𝑖 ∈ V ↦ ((freeMnd‘(𝑖 × 2𝑜)) /s ( ~FG ‘𝑖)))
12		ovex 6678	. . . 4 ⊢ (𝑀 /s ∼ ) ∈ V
13	10, 11, 12	fvmpt 6282	. . 3 ⊢ (𝐼 ∈ V → (freeGrp‘𝐼) = (𝑀 /s ∼ ))
14	2, 13	syl 17	. 2 ⊢ (𝐼 ∈ 𝑉 → (freeGrp‘𝐼) = (𝑀 /s ∼ ))
15	1, 14	syl5eq 2668	1 ⊢ (𝐼 ∈ 𝑉 → 𝐺 = (𝑀 /s ∼ ))

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  Vcvv 3200   × cxp 5112  ‘cfv 5888  (class class class)co 6650  2_𝑜c2o 7554   /_s cqus 16165  freeMndcfrmd 17384   ~_FG cefg 18119  freeGrpcfrgp 18120

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-frgp 18123

This theorem is referenced by:  frgp0  18173  frgpeccl  18174  frgpadd  18176  frgpupf  18186  frgpup1  18188  frgpup3lem  18190  frgpnabllem2  18277