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Theorem stafval 18842
 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 6189 . 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 18841 . 2 = (𝑥𝐵 ↦ ( 𝑥))
6 fvex 6199 . 2 ( 𝐴) ∈ V
71, 5, 6fvmpt 6280 1 (𝐴𝐵 → ( 𝐴) = ( 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1482   ∈ wcel 1989  ‘cfv 5886  Basecbs 15851  *𝑟cstv 15937  *rfcstf 18837 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-fv 5894  df-staf 18839 This theorem is referenced by:  srngcl  18849  srngnvl  18850  srngadd  18851  srngmul  18852  srng1  18853  srng0  18854  issrngd  18855  iporthcom  19974
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