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Theorem frgpval 19776
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 = ( ~FG𝐼)
Assertion
Ref Expression
frgpval (𝐼𝑉𝐺 = (𝑀 /s ))

Proof of Theorem frgpval
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 frgpval.m . 2 𝐺 = (freeGrp‘𝐼)
2 elex 3501 . . 3 (𝐼𝑉𝐼 ∈ V)
3 xpeq1 5699 . . . . . . 7 (𝑖 = 𝐼 → (𝑖 × 2o) = (𝐼 × 2o))
43fveq2d 6910 . . . . . 6 (𝑖 = 𝐼 → (freeMnd‘(𝑖 × 2o)) = (freeMnd‘(𝐼 × 2o)))
5 frgpval.b . . . . . 6 𝑀 = (freeMnd‘(𝐼 × 2o))
64, 5eqtr4di 2795 . . . . 5 (𝑖 = 𝐼 → (freeMnd‘(𝑖 × 2o)) = 𝑀)
7 fveq2 6906 . . . . . 6 (𝑖 = 𝐼 → ( ~FG𝑖) = ( ~FG𝐼))
8 frgpval.r . . . . . 6 = ( ~FG𝐼)
97, 8eqtr4di 2795 . . . . 5 (𝑖 = 𝐼 → ( ~FG𝑖) = )
106, 9oveq12d 7449 . . . 4 (𝑖 = 𝐼 → ((freeMnd‘(𝑖 × 2o)) /s ( ~FG𝑖)) = (𝑀 /s ))
11 df-frgp 19728 . . . 4 freeGrp = (𝑖 ∈ V ↦ ((freeMnd‘(𝑖 × 2o)) /s ( ~FG𝑖)))
12 ovex 7464 . . . 4 (𝑀 /s ) ∈ V
1310, 11, 12fvmpt 7016 . . 3 (𝐼 ∈ V → (freeGrp‘𝐼) = (𝑀 /s ))
142, 13syl 17 . 2 (𝐼𝑉 → (freeGrp‘𝐼) = (𝑀 /s ))
151, 14eqtrid 2789 1 (𝐼𝑉𝐺 = (𝑀 /s ))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480   × cxp 5683  cfv 6561  (class class class)co 7431  2oc2o 8500   /s cqus 17550  freeMndcfrmd 18860   ~FG cefg 19724  freeGrpcfrgp 19725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-frgp 19728
This theorem is referenced by:  frgp0  19778  frgpeccl  19779  frgpadd  19781  frgpupf  19791  frgpup1  19793  frgpup3lem  19795  frgpnabllem2  19892
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