Theorem frgpval 19828

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 3484	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V)
3		xpeq1 5676	. . . . . . 7 ⊢ (𝑖 = 𝐼 → (𝑖 × 2o) = (𝐼 × 2o))
4	3	fveq2d 6886	. . . . . 6 ⊢ (𝑖 = 𝐼 → (freeMnd‘(𝑖 × 2o)) = (freeMnd‘(𝐼 × 2o)))
5		frgpval.b	. . . . . 6 ⊢ 𝑀 = (freeMnd‘(𝐼 × 2o))
6	4, 5	eqtr4di 2822	. . . . 5 ⊢ (𝑖 = 𝐼 → (freeMnd‘(𝑖 × 2o)) = 𝑀)
7		fveq2 6882	. . . . . 6 ⊢ (𝑖 = 𝐼 → ( ~FG ‘𝑖) = ( ~FG ‘𝐼))
8		frgpval.r	. . . . . 6 ⊢ ∼ = ( ~FG ‘𝐼)
9	7, 8	eqtr4di 2822	. . . . 5 ⊢ (𝑖 = 𝐼 → ( ~FG ‘𝑖) = ∼ )
10	6, 9	oveq12d 7429	. . . 4 ⊢ (𝑖 = 𝐼 → ((freeMnd‘(𝑖 × 2o)) /s ( ~FG ‘𝑖)) = (𝑀 /s ∼ ))
11		df-frgp 19780	. . . 4 ⊢ freeGrp = (𝑖 ∈ V ↦ ((freeMnd‘(𝑖 × 2o)) /s ( ~FG ‘𝑖)))
12		ovex 7444	. . . 4 ⊢ (𝑀 /s ∼ ) ∈ V
13	10, 11, 12	fvmpt 6990	. . 3 ⊢ (𝐼 ∈ V → (freeGrp‘𝐼) = (𝑀 /s ∼ ))
14	2, 13	syl 18	. 2 ⊢ (𝐼 ∈ 𝑉 → (freeGrp‘𝐼) = (𝑀 /s ∼ ))
15	1, 14	eqtrid 2816	1 ⊢ (𝐼 ∈ 𝑉 → 𝐺 = (𝑀 /s ∼ ))

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  Vcvv 3463   × cxp 5660  ‘cfv 6537  (class class class)co 7411  2_oc2o 8447   /_s cqus 17559  freeMndcfrmd 18906   ~_FG cefg 19776  freeGrpcfrgp 19777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7414  df-frgp 19780

This theorem is referenced by:  frgp0  19830  frgpeccl  19831  frgpadd  19833  frgpupf  19843  frgpup1  19845  frgpup3lem  19847  frgpnabllem2  19944