Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
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 2824 | . 2 ⊢ (𝑔 = 𝐹 → ( I ↾ 𝐵) = ( I ↾ 𝐵)) | |
2 | tendo0cbv.o | . . 3 ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
3 | 2 | tendo0cbv 37924 | . 2 ⊢ 𝑂 = (𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵)) |
4 | funi 6389 | . . 3 ⊢ Fun I | |
5 | tendo02.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
6 | 5 | fvexi 6686 | . . 3 ⊢ 𝐵 ∈ V |
7 | resfunexg 6980 | . . 3 ⊢ ((Fun I ∧ 𝐵 ∈ V) → ( I ↾ 𝐵) ∈ V) | |
8 | 4, 6, 7 | mp2an 690 | . 2 ⊢ ( I ↾ 𝐵) ∈ V |
9 | 1, 3, 8 | fvmpt 6770 | 1 ⊢ (𝐹 ∈ 𝑇 → (𝑂‘𝐹) = ( I ↾ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ↦ cmpt 5148 I cid 5461 ↾ cres 5559 Fun wfun 6351 ‘cfv 6357 Basecbs 16485 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 |
This theorem is referenced by: tendo0co2 37926 tendo0tp 37927 tendo0pl 37929 tendoipl 37935 tendoid0 37963 tendo0mul 37964 tendo0mulr 37965 tendo1ne0 37966 tendoex 38113 dicn0 38330 dihordlem7b 38353 dihmeetlem1N 38428 dihglblem5apreN 38429 dihmeetlem4preN 38444 dihmeetlem13N 38457 |
Copyright terms: Public domain | W3C validator |