MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  stafval Structured version   Visualization version   GIF version

Theorem stafval 20860
Description: The functionalization of the involution component of a structure. (Contributed by Mario Carneiro, 6-Oct-2015.)
Hypotheses
Ref Expression
staffval.b 𝐵 = (Base‘𝑅)
staffval.i = (*𝑟𝑅)
staffval.f = (*rf𝑅)
Assertion
Ref Expression
stafval (𝐴𝐵 → ( 𝐴) = ( 𝐴))

Proof of Theorem stafval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6907 . 2 (𝑥 = 𝐴 → ( 𝑥) = ( 𝐴))
2 staffval.b . . 3 𝐵 = (Base‘𝑅)
3 staffval.i . . 3 = (*𝑟𝑅)
4 staffval.f . . 3 = (*rf𝑅)
52, 3, 4staffval 20859 . 2 = (𝑥𝐵 ↦ ( 𝑥))
6 fvex 6920 . 2 ( 𝐴) ∈ V
71, 5, 6fvmpt 7016 1 (𝐴𝐵 → ( 𝐴) = ( 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  cfv 6563  Basecbs 17245  *𝑟cstv 17300  *rfcstf 20855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-staf 20857
This theorem is referenced by:  srngcl  20867  srngnvl  20868  srngadd  20869  srngmul  20870  srng1  20871  srng0  20872  issrngd  20873  iporthcom  21671
  Copyright terms: Public domain W3C validator