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Theorem tendo02 36596
Description: Value of additive identity endomorphism. (Contributed by NM, 11-Jun-2013.)
Hypotheses
Ref Expression
tendo0cbv.o 𝑂 = (𝑓𝑇 ↦ ( I ↾ 𝐵))
tendo02.b 𝐵 = (Base‘𝐾)
Assertion
Ref Expression
tendo02 (𝐹𝑇 → (𝑂𝐹) = ( I ↾ 𝐵))
Distinct variable groups:   𝐵,𝑓   𝑇,𝑓
Allowed substitution hints:   𝐹(𝑓)   𝐾(𝑓)   𝑂(𝑓)

Proof of Theorem tendo02
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 eqidd 2772 . 2 (𝑔 = 𝐹 → ( I ↾ 𝐵) = ( I ↾ 𝐵))
2 tendo0cbv.o . . 3 𝑂 = (𝑓𝑇 ↦ ( I ↾ 𝐵))
32tendo0cbv 36595 . 2 𝑂 = (𝑔𝑇 ↦ ( I ↾ 𝐵))
4 funi 6063 . . 3 Fun I
5 tendo02.b . . . 4 𝐵 = (Base‘𝐾)
6 fvex 6342 . . . 4 (Base‘𝐾) ∈ V
75, 6eqeltri 2846 . . 3 𝐵 ∈ V
8 resfunexg 6623 . . 3 ((Fun I ∧ 𝐵 ∈ V) → ( I ↾ 𝐵) ∈ V)
94, 7, 8mp2an 672 . 2 ( I ↾ 𝐵) ∈ V
101, 3, 9fvmpt 6424 1 (𝐹𝑇 → (𝑂𝐹) = ( I ↾ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  Vcvv 3351  cmpt 4863   I cid 5156  cres 5251  Fun wfun 6025  cfv 6031  Basecbs 16064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039
This theorem is referenced by:  tendo0co2  36597  tendo0tp  36598  tendo0pl  36600  tendoipl  36606  tendoid0  36634  tendo0mul  36635  tendo0mulr  36636  tendo1ne0  36637  tendoex  36784  dicn0  37002  dihordlem7b  37025  dihmeetlem1N  37100  dihglblem5apreN  37101  dihmeetlem4preN  37116  dihmeetlem13N  37129
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