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Theorem frgpval 19675
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 3487 . . 3 (𝐼 ∈ 𝑉 β†’ 𝐼 ∈ V)
3 xpeq1 5683 . . . . . . 7 (𝑖 = 𝐼 β†’ (𝑖 Γ— 2o) = (𝐼 Γ— 2o))
43fveq2d 6888 . . . . . 6 (𝑖 = 𝐼 β†’ (freeMndβ€˜(𝑖 Γ— 2o)) = (freeMndβ€˜(𝐼 Γ— 2o)))
5 frgpval.b . . . . . 6 𝑀 = (freeMndβ€˜(𝐼 Γ— 2o))
64, 5eqtr4di 2784 . . . . 5 (𝑖 = 𝐼 β†’ (freeMndβ€˜(𝑖 Γ— 2o)) = 𝑀)
7 fveq2 6884 . . . . . 6 (𝑖 = 𝐼 β†’ ( ~FG β€˜π‘–) = ( ~FG β€˜πΌ))
8 frgpval.r . . . . . 6 ∼ = ( ~FG β€˜πΌ)
97, 8eqtr4di 2784 . . . . 5 (𝑖 = 𝐼 β†’ ( ~FG β€˜π‘–) = ∼ )
106, 9oveq12d 7422 . . . 4 (𝑖 = 𝐼 β†’ ((freeMndβ€˜(𝑖 Γ— 2o)) /s ( ~FG β€˜π‘–)) = (𝑀 /s ∼ ))
11 df-frgp 19627 . . . 4 freeGrp = (𝑖 ∈ V ↦ ((freeMndβ€˜(𝑖 Γ— 2o)) /s ( ~FG β€˜π‘–)))
12 ovex 7437 . . . 4 (𝑀 /s ∼ ) ∈ V
1310, 11, 12fvmpt 6991 . . 3 (𝐼 ∈ V β†’ (freeGrpβ€˜πΌ) = (𝑀 /s ∼ ))
142, 13syl 17 . 2 (𝐼 ∈ 𝑉 β†’ (freeGrpβ€˜πΌ) = (𝑀 /s ∼ ))
151, 14eqtrid 2778 1 (𝐼 ∈ 𝑉 β†’ 𝐺 = (𝑀 /s ∼ ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  Vcvv 3468   Γ— cxp 5667  β€˜cfv 6536  (class class class)co 7404  2oc2o 8458   /s cqus 17457  freeMndcfrmd 18769   ~FG cefg 19623  freeGrpcfrgp 19624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-ov 7407  df-frgp 19627
This theorem is referenced by:  frgp0  19677  frgpeccl  19678  frgpadd  19680  frgpupf  19690  frgpup1  19692  frgpup3lem  19694  frgpnabllem2  19791
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