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Theorem staffval 20842
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 6906 . . . . . 6 (𝑓 = 𝑅 → (Base‘𝑓) = (Base‘𝑅))
3 staffval.b . . . . . 6 𝐵 = (Base‘𝑅)
42, 3eqtr4di 2795 . . . . 5 (𝑓 = 𝑅 → (Base‘𝑓) = 𝐵)
5 fveq2 6906 . . . . . . 7 (𝑓 = 𝑅 → (*𝑟𝑓) = (*𝑟𝑅))
6 staffval.i . . . . . . 7 = (*𝑟𝑅)
75, 6eqtr4di 2795 . . . . . 6 (𝑓 = 𝑅 → (*𝑟𝑓) = )
87fveq1d 6908 . . . . 5 (𝑓 = 𝑅 → ((*𝑟𝑓)‘𝑥) = ( 𝑥))
94, 8mpteq12dv 5233 . . . 4 (𝑓 = 𝑅 → (𝑥 ∈ (Base‘𝑓) ↦ ((*𝑟𝑓)‘𝑥)) = (𝑥𝐵 ↦ ( 𝑥)))
10 df-staf 20840 . . . 4 *rf = (𝑓 ∈ V ↦ (𝑥 ∈ (Base‘𝑓) ↦ ((*𝑟𝑓)‘𝑥)))
11 eqid 2737 . . . . . 6 (𝑥𝐵 ↦ ( 𝑥)) = (𝑥𝐵 ↦ ( 𝑥))
12 fvrn0 6936 . . . . . . 7 ( 𝑥) ∈ (ran ∪ {∅})
1312a1i 11 . . . . . 6 (𝑥𝐵 → ( 𝑥) ∈ (ran ∪ {∅}))
1411, 13fmpti 7132 . . . . 5 (𝑥𝐵 ↦ ( 𝑥)):𝐵⟶(ran ∪ {∅})
153fvexi 6920 . . . . 5 𝐵 ∈ V
166fvexi 6920 . . . . . . 7 ∈ V
1716rnex 7932 . . . . . 6 ran ∈ V
18 p0ex 5384 . . . . . 6 {∅} ∈ V
1917, 18unex 7764 . . . . 5 (ran ∪ {∅}) ∈ V
20 fex2 7958 . . . . 5 (((𝑥𝐵 ↦ ( 𝑥)):𝐵⟶(ran ∪ {∅}) ∧ 𝐵 ∈ V ∧ (ran ∪ {∅}) ∈ V) → (𝑥𝐵 ↦ ( 𝑥)) ∈ V)
2114, 15, 19, 20mp3an 1463 . . . 4 (𝑥𝐵 ↦ ( 𝑥)) ∈ V
229, 10, 21fvmpt 7016 . . 3 (𝑅 ∈ V → (*rf𝑅) = (𝑥𝐵 ↦ ( 𝑥)))
23 fvprc 6898 . . . . 5 𝑅 ∈ V → (*rf𝑅) = ∅)
24 mpt0 6710 . . . . 5 (𝑥 ∈ ∅ ↦ ( 𝑥)) = ∅
2523, 24eqtr4di 2795 . . . 4 𝑅 ∈ V → (*rf𝑅) = (𝑥 ∈ ∅ ↦ ( 𝑥)))
26 fvprc 6898 . . . . . 6 𝑅 ∈ V → (Base‘𝑅) = ∅)
273, 26eqtrid 2789 . . . . 5 𝑅 ∈ V → 𝐵 = ∅)
2827mpteq1d 5237 . . . 4 𝑅 ∈ V → (𝑥𝐵 ↦ ( 𝑥)) = (𝑥 ∈ ∅ ↦ ( 𝑥)))
2925, 28eqtr4d 2780 . . 3 𝑅 ∈ V → (*rf𝑅) = (𝑥𝐵 ↦ ( 𝑥)))
3022, 29pm2.61i 182 . 2 (*rf𝑅) = (𝑥𝐵 ↦ ( 𝑥))
311, 30eqtri 2765 1 = (𝑥𝐵 ↦ ( 𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1540  wcel 2108  Vcvv 3480  cun 3949  c0 4333  {csn 4626  cmpt 5225  ran crn 5686  wf 6557  cfv 6561  Basecbs 17247  *𝑟cstv 17299  *rfcstf 20838
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-staf 20840
This theorem is referenced by:  stafval  20843  staffn  20844  issrngd  20856
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