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Mirrors > Home > MPE Home > Th. List > Mathboxes > tendo02 | Structured version Visualization version GIF version |
Description: Value of additive identity endomorphism. (Contributed by NM, 11-Jun-2013.) |
Ref | Expression |
---|---|
tendo0cbv.o | ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) |
tendo02.b | ⊢ 𝐵 = (Base‘𝐾) |
Ref | Expression |
---|---|
tendo02 | ⊢ (𝐹 ∈ 𝑇 → (𝑂‘𝐹) = ( I ↾ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2826 | . 2 ⊢ (𝑔 = 𝐹 → ( I ↾ 𝐵) = ( I ↾ 𝐵)) | |
2 | tendo0cbv.o | . . 3 ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
3 | 2 | tendo0cbv 36856 | . 2 ⊢ 𝑂 = (𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵)) |
4 | funi 6159 | . . 3 ⊢ Fun I | |
5 | tendo02.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
6 | 5 | fvexi 6451 | . . 3 ⊢ 𝐵 ∈ V |
7 | resfunexg 6740 | . . 3 ⊢ ((Fun I ∧ 𝐵 ∈ V) → ( I ↾ 𝐵) ∈ V) | |
8 | 4, 6, 7 | mp2an 683 | . 2 ⊢ ( I ↾ 𝐵) ∈ V |
9 | 1, 3, 8 | fvmpt 6533 | 1 ⊢ (𝐹 ∈ 𝑇 → (𝑂‘𝐹) = ( I ↾ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 Vcvv 3414 ↦ cmpt 4954 I cid 5251 ↾ cres 5348 Fun wfun 6121 ‘cfv 6127 Basecbs 16229 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 |
This theorem is referenced by: tendo0co2 36858 tendo0tp 36859 tendo0pl 36861 tendoipl 36867 tendoid0 36895 tendo0mul 36896 tendo0mulr 36897 tendo1ne0 36898 tendoex 37045 dicn0 37262 dihordlem7b 37285 dihmeetlem1N 37360 dihglblem5apreN 37361 dihmeetlem4preN 37376 dihmeetlem13N 37389 |
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