Theorem frgpval 18886

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 3514	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V)
3		xpeq1 5571	. . . . . . 7 ⊢ (𝑖 = 𝐼 → (𝑖 × 2o) = (𝐼 × 2o))
4	3	fveq2d 6676	. . . . . 6 ⊢ (𝑖 = 𝐼 → (freeMnd‘(𝑖 × 2o)) = (freeMnd‘(𝐼 × 2o)))
5		frgpval.b	. . . . . 6 ⊢ 𝑀 = (freeMnd‘(𝐼 × 2o))
6	4, 5	syl6eqr 2876	. . . . 5 ⊢ (𝑖 = 𝐼 → (freeMnd‘(𝑖 × 2o)) = 𝑀)
7		fveq2 6672	. . . . . 6 ⊢ (𝑖 = 𝐼 → ( ~FG ‘𝑖) = ( ~FG ‘𝐼))
8		frgpval.r	. . . . . 6 ⊢ ∼ = ( ~FG ‘𝐼)
9	7, 8	syl6eqr 2876	. . . . 5 ⊢ (𝑖 = 𝐼 → ( ~FG ‘𝑖) = ∼ )
10	6, 9	oveq12d 7176	. . . 4 ⊢ (𝑖 = 𝐼 → ((freeMnd‘(𝑖 × 2o)) /s ( ~FG ‘𝑖)) = (𝑀 /s ∼ ))
11		df-frgp 18838	. . . 4 ⊢ freeGrp = (𝑖 ∈ V ↦ ((freeMnd‘(𝑖 × 2o)) /s ( ~FG ‘𝑖)))
12		ovex 7191	. . . 4 ⊢ (𝑀 /s ∼ ) ∈ V
13	10, 11, 12	fvmpt 6770	. . 3 ⊢ (𝐼 ∈ V → (freeGrp‘𝐼) = (𝑀 /s ∼ ))
14	2, 13	syl 17	. 2 ⊢ (𝐼 ∈ 𝑉 → (freeGrp‘𝐼) = (𝑀 /s ∼ ))
15	1, 14	syl5eq 2870	1 ⊢ (𝐼 ∈ 𝑉 → 𝐺 = (𝑀 /s ∼ ))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  Vcvv 3496   × cxp 5555  ‘cfv 6357  (class class class)co 7158  2_oc2o 8098   /_s cqus 16780  freeMndcfrmd 18014   ~_FG cefg 18834  freeGrpcfrgp 18835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-iota 6316  df-fun 6359  df-fv 6365  df-ov 7161  df-frgp 18838

This theorem is referenced by:  frgp0  18888  frgpeccl  18889  frgpadd  18891  frgpupf  18901  frgpup1  18903  frgpup3lem  18905  frgpnabllem2  18996