Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 3185 | . . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V) | |
3 | id 22 | . . . . 5 ⊢ (𝑖 = 𝐼 → 𝑖 = 𝐼) | |
4 | fveq2 6103 | . . . . . . 7 ⊢ (𝑖 = 𝐼 → ( ~FG ‘𝑖) = ( ~FG ‘𝐼)) | |
5 | vrgpfval.r | . . . . . . 7 ⊢ ∼ = ( ~FG ‘𝐼) | |
6 | 4, 5 | syl6eqr 2662 | . . . . . 6 ⊢ (𝑖 = 𝐼 → ( ~FG ‘𝑖) = ∼ ) |
7 | eceq2 7671 | . . . . . 6 ⊢ (( ~FG ‘𝑖) = ∼ → [〈“〈𝑗, ∅〉”〉]( ~FG ‘𝑖) = [〈“〈𝑗, ∅〉”〉] ∼ ) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ (𝑖 = 𝐼 → [〈“〈𝑗, ∅〉”〉]( ~FG ‘𝑖) = [〈“〈𝑗, ∅〉”〉] ∼ ) |
9 | 3, 8 | mpteq12dv 4663 | . . . 4 ⊢ (𝑖 = 𝐼 → (𝑗 ∈ 𝑖 ↦ [〈“〈𝑗, ∅〉”〉]( ~FG ‘𝑖)) = (𝑗 ∈ 𝐼 ↦ [〈“〈𝑗, ∅〉”〉] ∼ )) |
10 | df-vrgp 17947 | . . . 4 ⊢ varFGrp = (𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ [〈“〈𝑗, ∅〉”〉]( ~FG ‘𝑖))) | |
11 | vex 3176 | . . . . 5 ⊢ 𝑖 ∈ V | |
12 | 11 | mptex 6390 | . . . 4 ⊢ (𝑗 ∈ 𝑖 ↦ [〈“〈𝑗, ∅〉”〉]( ~FG ‘𝑖)) ∈ V |
13 | 9, 10, 12 | fvmpt3i 6196 | . . 3 ⊢ (𝐼 ∈ V → (varFGrp‘𝐼) = (𝑗 ∈ 𝐼 ↦ [〈“〈𝑗, ∅〉”〉] ∼ )) |
14 | 2, 13 | syl 17 | . 2 ⊢ (𝐼 ∈ 𝑉 → (varFGrp‘𝐼) = (𝑗 ∈ 𝐼 ↦ [〈“〈𝑗, ∅〉”〉] ∼ )) |
15 | 1, 14 | syl5eq 2656 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝑈 = (𝑗 ∈ 𝐼 ↦ [〈“〈𝑗, ∅〉”〉] ∼ )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 〈cop 4131 ↦ cmpt 4643 ‘cfv 5804 [cec 7627 〈“cs1 13149 ~FG cefg 17942 varFGrpcvrgp 17944 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ec 7631 df-vrgp 17947 |
This theorem is referenced by: vrgpval 18003 vrgpf 18004 |
Copyright terms: Public domain | W3C validator |