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Theorem frgpval 19791
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 3499 . . 3 (𝐼𝑉𝐼 ∈ V)
3 xpeq1 5703 . . . . . . 7 (𝑖 = 𝐼 → (𝑖 × 2o) = (𝐼 × 2o))
43fveq2d 6911 . . . . . 6 (𝑖 = 𝐼 → (freeMnd‘(𝑖 × 2o)) = (freeMnd‘(𝐼 × 2o)))
5 frgpval.b . . . . . 6 𝑀 = (freeMnd‘(𝐼 × 2o))
64, 5eqtr4di 2793 . . . . 5 (𝑖 = 𝐼 → (freeMnd‘(𝑖 × 2o)) = 𝑀)
7 fveq2 6907 . . . . . 6 (𝑖 = 𝐼 → ( ~FG𝑖) = ( ~FG𝐼))
8 frgpval.r . . . . . 6 = ( ~FG𝐼)
97, 8eqtr4di 2793 . . . . 5 (𝑖 = 𝐼 → ( ~FG𝑖) = )
106, 9oveq12d 7449 . . . 4 (𝑖 = 𝐼 → ((freeMnd‘(𝑖 × 2o)) /s ( ~FG𝑖)) = (𝑀 /s ))
11 df-frgp 19743 . . . 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 2787 1 (𝐼𝑉𝐺 = (𝑀 /s ))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  Vcvv 3478   × cxp 5687  cfv 6563  (class class class)co 7431  2oc2o 8499   /s cqus 17552  freeMndcfrmd 18873   ~FG cefg 19739  freeGrpcfrgp 19740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-frgp 19743
This theorem is referenced by:  frgp0  19793  frgpeccl  19794  frgpadd  19796  frgpupf  19806  frgpup1  19808  frgpup3lem  19810  frgpnabllem2  19907
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