Theorem frgpval 19625

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 3492	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V)
3		xpeq1 5690	. . . . . . 7 ⊢ (𝑖 = 𝐼 → (𝑖 × 2o) = (𝐼 × 2o))
4	3	fveq2d 6895	. . . . . 6 ⊢ (𝑖 = 𝐼 → (freeMnd‘(𝑖 × 2o)) = (freeMnd‘(𝐼 × 2o)))
5		frgpval.b	. . . . . 6 ⊢ 𝑀 = (freeMnd‘(𝐼 × 2o))
6	4, 5	eqtr4di 2790	. . . . 5 ⊢ (𝑖 = 𝐼 → (freeMnd‘(𝑖 × 2o)) = 𝑀)
7		fveq2 6891	. . . . . 6 ⊢ (𝑖 = 𝐼 → ( ~FG ‘𝑖) = ( ~FG ‘𝐼))
8		frgpval.r	. . . . . 6 ⊢ ∼ = ( ~FG ‘𝐼)
9	7, 8	eqtr4di 2790	. . . . 5 ⊢ (𝑖 = 𝐼 → ( ~FG ‘𝑖) = ∼ )
10	6, 9	oveq12d 7426	. . . 4 ⊢ (𝑖 = 𝐼 → ((freeMnd‘(𝑖 × 2o)) /s ( ~FG ‘𝑖)) = (𝑀 /s ∼ ))
11		df-frgp 19577	. . . 4 ⊢ freeGrp = (𝑖 ∈ V ↦ ((freeMnd‘(𝑖 × 2o)) /s ( ~FG ‘𝑖)))
12		ovex 7441	. . . 4 ⊢ (𝑀 /s ∼ ) ∈ V
13	10, 11, 12	fvmpt 6998	. . 3 ⊢ (𝐼 ∈ V → (freeGrp‘𝐼) = (𝑀 /s ∼ ))
14	2, 13	syl 17	. 2 ⊢ (𝐼 ∈ 𝑉 → (freeGrp‘𝐼) = (𝑀 /s ∼ ))
15	1, 14	eqtrid 2784	1 ⊢ (𝐼 ∈ 𝑉 → 𝐺 = (𝑀 /s ∼ ))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106  Vcvv 3474   × cxp 5674  ‘cfv 6543  (class class class)co 7408  2_oc2o 8459   /_s cqus 17450  freeMndcfrmd 18727   ~_FG cefg 19573  freeGrpcfrgp 19574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7411  df-frgp 19577

This theorem is referenced by:  frgp0  19627  frgpeccl  19628  frgpadd  19630  frgpupf  19640  frgpup1  19642  frgpup3lem  19644  frgpnabllem2  19741