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Theorem frgpval 19547
Description: Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-2015.)
Hypotheses
Ref Expression
frgpval.m 𝐺 = (freeGrpβ€˜πΌ)
frgpval.b 𝑀 = (freeMndβ€˜(𝐼 Γ— 2o))
frgpval.r ∼ = ( ~FG β€˜πΌ)
Assertion
Ref Expression
frgpval (𝐼 ∈ 𝑉 β†’ 𝐺 = (𝑀 /s ∼ ))

Proof of Theorem frgpval
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 frgpval.m . 2 𝐺 = (freeGrpβ€˜πΌ)
2 elex 3466 . . 3 (𝐼 ∈ 𝑉 β†’ 𝐼 ∈ V)
3 xpeq1 5652 . . . . . . 7 (𝑖 = 𝐼 β†’ (𝑖 Γ— 2o) = (𝐼 Γ— 2o))
43fveq2d 6851 . . . . . 6 (𝑖 = 𝐼 β†’ (freeMndβ€˜(𝑖 Γ— 2o)) = (freeMndβ€˜(𝐼 Γ— 2o)))
5 frgpval.b . . . . . 6 𝑀 = (freeMndβ€˜(𝐼 Γ— 2o))
64, 5eqtr4di 2795 . . . . 5 (𝑖 = 𝐼 β†’ (freeMndβ€˜(𝑖 Γ— 2o)) = 𝑀)
7 fveq2 6847 . . . . . 6 (𝑖 = 𝐼 β†’ ( ~FG β€˜π‘–) = ( ~FG β€˜πΌ))
8 frgpval.r . . . . . 6 ∼ = ( ~FG β€˜πΌ)
97, 8eqtr4di 2795 . . . . 5 (𝑖 = 𝐼 β†’ ( ~FG β€˜π‘–) = ∼ )
106, 9oveq12d 7380 . . . 4 (𝑖 = 𝐼 β†’ ((freeMndβ€˜(𝑖 Γ— 2o)) /s ( ~FG β€˜π‘–)) = (𝑀 /s ∼ ))
11 df-frgp 19499 . . . 4 freeGrp = (𝑖 ∈ V ↦ ((freeMndβ€˜(𝑖 Γ— 2o)) /s ( ~FG β€˜π‘–)))
12 ovex 7395 . . . 4 (𝑀 /s ∼ ) ∈ V
1310, 11, 12fvmpt 6953 . . 3 (𝐼 ∈ V β†’ (freeGrpβ€˜πΌ) = (𝑀 /s ∼ ))
142, 13syl 17 . 2 (𝐼 ∈ 𝑉 β†’ (freeGrpβ€˜πΌ) = (𝑀 /s ∼ ))
151, 14eqtrid 2789 1 (𝐼 ∈ 𝑉 β†’ 𝐺 = (𝑀 /s ∼ ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  Vcvv 3448   Γ— cxp 5636  β€˜cfv 6501  (class class class)co 7362  2oc2o 8411   /s cqus 17394  freeMndcfrmd 18664   ~FG cefg 19495  freeGrpcfrgp 19496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-ov 7365  df-frgp 19499
This theorem is referenced by:  frgp0  19549  frgpeccl  19550  frgpadd  19552  frgpupf  19562  frgpup1  19564  frgpup3lem  19566  frgpnabllem2  19659
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