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Theorem staffval 20859
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
staffval = (𝑥𝐵 ↦ ( 𝑥))
Distinct variable groups:   𝑥,𝐵   𝑥,   𝑥,𝑅
Allowed substitution hint:   (𝑥)

Proof of Theorem staffval
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 staffval.f . 2 = (*rf𝑅)
2 fveq2 6907 . . . . . 6 (𝑓 = 𝑅 → (Base‘𝑓) = (Base‘𝑅))
3 staffval.b . . . . . 6 𝐵 = (Base‘𝑅)
42, 3eqtr4di 2793 . . . . 5 (𝑓 = 𝑅 → (Base‘𝑓) = 𝐵)
5 fveq2 6907 . . . . . . 7 (𝑓 = 𝑅 → (*𝑟𝑓) = (*𝑟𝑅))
6 staffval.i . . . . . . 7 = (*𝑟𝑅)
75, 6eqtr4di 2793 . . . . . 6 (𝑓 = 𝑅 → (*𝑟𝑓) = )
87fveq1d 6909 . . . . 5 (𝑓 = 𝑅 → ((*𝑟𝑓)‘𝑥) = ( 𝑥))
94, 8mpteq12dv 5239 . . . 4 (𝑓 = 𝑅 → (𝑥 ∈ (Base‘𝑓) ↦ ((*𝑟𝑓)‘𝑥)) = (𝑥𝐵 ↦ ( 𝑥)))
10 df-staf 20857 . . . 4 *rf = (𝑓 ∈ V ↦ (𝑥 ∈ (Base‘𝑓) ↦ ((*𝑟𝑓)‘𝑥)))
11 eqid 2735 . . . . . 6 (𝑥𝐵 ↦ ( 𝑥)) = (𝑥𝐵 ↦ ( 𝑥))
12 fvrn0 6937 . . . . . . 7 ( 𝑥) ∈ (ran ∪ {∅})
1312a1i 11 . . . . . 6 (𝑥𝐵 → ( 𝑥) ∈ (ran ∪ {∅}))
1411, 13fmpti 7132 . . . . 5 (𝑥𝐵 ↦ ( 𝑥)):𝐵⟶(ran ∪ {∅})
153fvexi 6921 . . . . 5 𝐵 ∈ V
166fvexi 6921 . . . . . . 7 ∈ V
1716rnex 7933 . . . . . 6 ran ∈ V
18 p0ex 5390 . . . . . 6 {∅} ∈ V
1917, 18unex 7763 . . . . 5 (ran ∪ {∅}) ∈ V
20 fex2 7957 . . . . 5 (((𝑥𝐵 ↦ ( 𝑥)):𝐵⟶(ran ∪ {∅}) ∧ 𝐵 ∈ V ∧ (ran ∪ {∅}) ∈ V) → (𝑥𝐵 ↦ ( 𝑥)) ∈ V)
2114, 15, 19, 20mp3an 1460 . . . 4 (𝑥𝐵 ↦ ( 𝑥)) ∈ V
229, 10, 21fvmpt 7016 . . 3 (𝑅 ∈ V → (*rf𝑅) = (𝑥𝐵 ↦ ( 𝑥)))
23 fvprc 6899 . . . . 5 𝑅 ∈ V → (*rf𝑅) = ∅)
24 mpt0 6711 . . . . 5 (𝑥 ∈ ∅ ↦ ( 𝑥)) = ∅
2523, 24eqtr4di 2793 . . . 4 𝑅 ∈ V → (*rf𝑅) = (𝑥 ∈ ∅ ↦ ( 𝑥)))
26 fvprc 6899 . . . . . 6 𝑅 ∈ V → (Base‘𝑅) = ∅)
273, 26eqtrid 2787 . . . . 5 𝑅 ∈ V → 𝐵 = ∅)
2827mpteq1d 5243 . . . 4 𝑅 ∈ V → (𝑥𝐵 ↦ ( 𝑥)) = (𝑥 ∈ ∅ ↦ ( 𝑥)))
2925, 28eqtr4d 2778 . . 3 𝑅 ∈ V → (*rf𝑅) = (𝑥𝐵 ↦ ( 𝑥)))
3022, 29pm2.61i 182 . 2 (*rf𝑅) = (𝑥𝐵 ↦ ( 𝑥))
311, 30eqtri 2763 1 = (𝑥𝐵 ↦ ( 𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1537  wcel 2106  Vcvv 3478  cun 3961  c0 4339  {csn 4631  cmpt 5231  ran crn 5690  wf 6559  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:  stafval  20860  staffn  20861  issrngd  20873
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