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| Mirrors > Home > MPE Home > Th. List > vrgpval | Structured version Visualization version GIF version | ||
| Description: The value of the generating elements of a free group. (Contributed by Mario Carneiro, 2-Oct-2015.) | 
| Ref | Expression | 
|---|---|
| vrgpfval.r | ⊢ ∼ = ( ~FG ‘𝐼) | 
| vrgpfval.u | ⊢ 𝑈 = (varFGrp‘𝐼) | 
| Ref | Expression | 
|---|---|
| vrgpval | ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑈‘𝐴) = [〈“〈𝐴, ∅〉”〉] ∼ ) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | vrgpfval.r | . . . 4 ⊢ ∼ = ( ~FG ‘𝐼) | |
| 2 | vrgpfval.u | . . . 4 ⊢ 𝑈 = (varFGrp‘𝐼) | |
| 3 | 1, 2 | vrgpfval 19785 | . . 3 ⊢ (𝐼 ∈ 𝑉 → 𝑈 = (𝑗 ∈ 𝐼 ↦ [〈“〈𝑗, ∅〉”〉] ∼ )) | 
| 4 | 3 | fveq1d 6907 | . 2 ⊢ (𝐼 ∈ 𝑉 → (𝑈‘𝐴) = ((𝑗 ∈ 𝐼 ↦ [〈“〈𝑗, ∅〉”〉] ∼ )‘𝐴)) | 
| 5 | opeq1 4872 | . . . . 5 ⊢ (𝑗 = 𝐴 → 〈𝑗, ∅〉 = 〈𝐴, ∅〉) | |
| 6 | 5 | s1eqd 14640 | . . . 4 ⊢ (𝑗 = 𝐴 → 〈“〈𝑗, ∅〉”〉 = 〈“〈𝐴, ∅〉”〉) | 
| 7 | 6 | eceq1d 8786 | . . 3 ⊢ (𝑗 = 𝐴 → [〈“〈𝑗, ∅〉”〉] ∼ = [〈“〈𝐴, ∅〉”〉] ∼ ) | 
| 8 | eqid 2736 | . . 3 ⊢ (𝑗 ∈ 𝐼 ↦ [〈“〈𝑗, ∅〉”〉] ∼ ) = (𝑗 ∈ 𝐼 ↦ [〈“〈𝑗, ∅〉”〉] ∼ ) | |
| 9 | 1 | fvexi 6919 | . . . 4 ⊢ ∼ ∈ V | 
| 10 | ecexg 8750 | . . . 4 ⊢ ( ∼ ∈ V → [〈“〈𝐴, ∅〉”〉] ∼ ∈ V) | |
| 11 | 9, 10 | ax-mp 5 | . . 3 ⊢ [〈“〈𝐴, ∅〉”〉] ∼ ∈ V | 
| 12 | 7, 8, 11 | fvmpt 7015 | . 2 ⊢ (𝐴 ∈ 𝐼 → ((𝑗 ∈ 𝐼 ↦ [〈“〈𝑗, ∅〉”〉] ∼ )‘𝐴) = [〈“〈𝐴, ∅〉”〉] ∼ ) | 
| 13 | 4, 12 | sylan9eq 2796 | 1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑈‘𝐴) = [〈“〈𝐴, ∅〉”〉] ∼ ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ∅c0 4332 〈cop 4631 ↦ cmpt 5224 ‘cfv 6560 [cec 8744 〈“cs1 14634 ~FG cefg 19725 varFGrpcvrgp 19727 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ec 8748 df-s1 14635 df-vrgp 19730 | 
| This theorem is referenced by: vrgpinv 19788 frgpup2 19795 frgpup3lem 19796 frgpnabllem1 19892 | 
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