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Theorem frgpval 19499
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 3461 . . 3 (𝐼𝑉𝐼 ∈ V)
3 xpeq1 5645 . . . . . . 7 (𝑖 = 𝐼 → (𝑖 × 2o) = (𝐼 × 2o))
43fveq2d 6843 . . . . . 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 6839 . . . . . 6 (𝑖 = 𝐼 → ( ~FG𝑖) = ( ~FG𝐼))
8 frgpval.r . . . . . 6 = ( ~FG𝐼)
97, 8eqtr4di 2795 . . . . 5 (𝑖 = 𝐼 → ( ~FG𝑖) = )
106, 9oveq12d 7369 . . . 4 (𝑖 = 𝐼 → ((freeMnd‘(𝑖 × 2o)) /s ( ~FG𝑖)) = (𝑀 /s ))
11 df-frgp 19451 . . . 4 freeGrp = (𝑖 ∈ V ↦ ((freeMnd‘(𝑖 × 2o)) /s ( ~FG𝑖)))
12 ovex 7384 . . . 4 (𝑀 /s ) ∈ V
1310, 11, 12fvmpt 6945 . . 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 1541  wcel 2106  Vcvv 3443   × cxp 5629  cfv 6493  (class class class)co 7351  2oc2o 8398   /s cqus 17347  freeMndcfrmd 18617   ~FG cefg 19447  freeGrpcfrgp 19448
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 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6445  df-fun 6495  df-fv 6501  df-ov 7354  df-frgp 19451
This theorem is referenced by:  frgp0  19501  frgpeccl  19502  frgpadd  19504  frgpupf  19514  frgpup1  19516  frgpup3lem  19518  frgpnabllem2  19611
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