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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 18015 | . . 3 ⊢ varFMnd = (𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ 〈“𝑗”〉)) | |
3 | mpteq1 5154 | . . 3 ⊢ (𝑖 = 𝐼 → (𝑗 ∈ 𝑖 ↦ 〈“𝑗”〉) = (𝑗 ∈ 𝐼 ↦ 〈“𝑗”〉)) | |
4 | elex 3512 | . . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V) | |
5 | mptexg 6984 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (𝑗 ∈ 𝐼 ↦ 〈“𝑗”〉) ∈ V) | |
6 | 2, 3, 4, 5 | fvmptd3 6791 | . 2 ⊢ (𝐼 ∈ 𝑉 → (varFMnd‘𝐼) = (𝑗 ∈ 𝐼 ↦ 〈“𝑗”〉)) |
7 | 1, 6 | syl5eq 2868 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝑈 = (𝑗 ∈ 𝐼 ↦ 〈“𝑗”〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ↦ cmpt 5146 ‘cfv 6355 〈“cs1 13949 varFMndcvrmd 18013 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-vrmd 18015 |
This theorem is referenced by: vrmdval 18022 vrmdf 18023 frgpup3lem 18903 |
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