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Theorem stafval 20758
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 6861 . 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 20757 . 2 = (𝑥𝐵 ↦ ( 𝑥))
6 fvex 6874 . 2 ( 𝐴) ∈ V
71, 5, 6fvmpt 6971 1 (𝐴𝐵 → ( 𝐴) = ( 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2109  cfv 6514  Basecbs 17186  *𝑟cstv 17229  *rfcstf 20753
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-staf 20755
This theorem is referenced by:  srngcl  20765  srngnvl  20766  srngadd  20767  srngmul  20768  srng1  20769  srng0  20770  issrngd  20771  iporthcom  21551
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