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| Mirrors > Home > MPE Home > Th. List > vrgpfval | Structured version Visualization version GIF version | ||
| Description: The canonical injection from the generating set 𝐼 to the base set of the free group. (Contributed by Mario Carneiro, 2-Oct-2015.) |
| Ref | Expression |
|---|---|
| vrgpfval.r | ⊢ ∼ = ( ~FG ‘𝐼) |
| vrgpfval.u | ⊢ 𝑈 = (varFGrp‘𝐼) |
| Ref | Expression |
|---|---|
| vrgpfval | ⊢ (𝐼 ∈ 𝑉 → 𝑈 = (𝑗 ∈ 𝐼 ↦ [〈“〈𝑗, ∅〉”〉] ∼ )) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | vrgpfval.u | . 2 ⊢ 𝑈 = (varFGrp‘𝐼) | |
| 2 | elex 3501 | . . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V) | |
| 3 | id 22 | . . . . 5 ⊢ (𝑖 = 𝐼 → 𝑖 = 𝐼) | |
| 4 | fveq2 6906 | . . . . . . 7 ⊢ (𝑖 = 𝐼 → ( ~FG ‘𝑖) = ( ~FG ‘𝐼)) | |
| 5 | vrgpfval.r | . . . . . . 7 ⊢ ∼ = ( ~FG ‘𝐼) | |
| 6 | 4, 5 | eqtr4di 2795 | . . . . . 6 ⊢ (𝑖 = 𝐼 → ( ~FG ‘𝑖) = ∼ ) |
| 7 | 6 | eceq2d 8788 | . . . . 5 ⊢ (𝑖 = 𝐼 → [〈“〈𝑗, ∅〉”〉]( ~FG ‘𝑖) = [〈“〈𝑗, ∅〉”〉] ∼ ) |
| 8 | 3, 7 | mpteq12dv 5233 | . . . 4 ⊢ (𝑖 = 𝐼 → (𝑗 ∈ 𝑖 ↦ [〈“〈𝑗, ∅〉”〉]( ~FG ‘𝑖)) = (𝑗 ∈ 𝐼 ↦ [〈“〈𝑗, ∅〉”〉] ∼ )) |
| 9 | df-vrgp 19729 | . . . 4 ⊢ varFGrp = (𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ [〈“〈𝑗, ∅〉”〉]( ~FG ‘𝑖))) | |
| 10 | vex 3484 | . . . . 5 ⊢ 𝑖 ∈ V | |
| 11 | 10 | mptex 7243 | . . . 4 ⊢ (𝑗 ∈ 𝑖 ↦ [〈“〈𝑗, ∅〉”〉]( ~FG ‘𝑖)) ∈ V |
| 12 | 8, 9, 11 | fvmpt3i 7021 | . . 3 ⊢ (𝐼 ∈ V → (varFGrp‘𝐼) = (𝑗 ∈ 𝐼 ↦ [〈“〈𝑗, ∅〉”〉] ∼ )) |
| 13 | 2, 12 | syl 17 | . 2 ⊢ (𝐼 ∈ 𝑉 → (varFGrp‘𝐼) = (𝑗 ∈ 𝐼 ↦ [〈“〈𝑗, ∅〉”〉] ∼ )) |
| 14 | 1, 13 | eqtrid 2789 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝑈 = (𝑗 ∈ 𝐼 ↦ [〈“〈𝑗, ∅〉”〉] ∼ )) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 〈cop 4632 ↦ cmpt 5225 ‘cfv 6561 [cec 8743 〈“cs1 14633 ~FG cefg 19724 varFGrpcvrgp 19726 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ec 8747 df-vrgp 19729 |
| This theorem is referenced by: vrgpval 19785 vrgpf 19786 |
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