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Mirrors > Home > MPE Home > Th. List > vrmdfval | Structured version Visualization version GIF version |
Description: The canonical injection from the generating set 𝐼 to the base set of the free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.) |
Ref | Expression |
---|---|
vrmdfval.u | ⊢ 𝑈 = (varFMnd‘𝐼) |
Ref | Expression |
---|---|
vrmdfval | ⊢ (𝐼 ∈ 𝑉 → 𝑈 = (𝑗 ∈ 𝐼 ↦ ⟨“𝑗”⟩)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vrmdfval.u | . 2 ⊢ 𝑈 = (varFMnd‘𝐼) | |
2 | df-vrmd 18809 | . . 3 ⊢ varFMnd = (𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ ⟨“𝑗”⟩)) | |
3 | mpteq1 5245 | . . 3 ⊢ (𝑖 = 𝐼 → (𝑗 ∈ 𝑖 ↦ ⟨“𝑗”⟩) = (𝑗 ∈ 𝐼 ↦ ⟨“𝑗”⟩)) | |
4 | elex 3492 | . . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V) | |
5 | mptexg 7239 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (𝑗 ∈ 𝐼 ↦ ⟨“𝑗”⟩) ∈ V) | |
6 | 2, 3, 4, 5 | fvmptd3 7033 | . 2 ⊢ (𝐼 ∈ 𝑉 → (varFMnd‘𝐼) = (𝑗 ∈ 𝐼 ↦ ⟨“𝑗”⟩)) |
7 | 1, 6 | eqtrid 2780 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝑈 = (𝑗 ∈ 𝐼 ↦ ⟨“𝑗”⟩)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3473 ↦ cmpt 5235 ‘cfv 6553 ⟨“cs1 14585 varFMndcvrmd 18807 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-vrmd 18809 |
This theorem is referenced by: vrmdval 18816 vrmdf 18817 frgpup3lem 19739 |
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