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Theorem staffval 18763
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 6150 . . . . . 6 (𝑓 = 𝑅 → (Base‘𝑓) = (Base‘𝑅))
3 staffval.b . . . . . 6 𝐵 = (Base‘𝑅)
42, 3syl6eqr 2678 . . . . 5 (𝑓 = 𝑅 → (Base‘𝑓) = 𝐵)
5 fveq2 6150 . . . . . . 7 (𝑓 = 𝑅 → (*𝑟𝑓) = (*𝑟𝑅))
6 staffval.i . . . . . . 7 = (*𝑟𝑅)
75, 6syl6eqr 2678 . . . . . 6 (𝑓 = 𝑅 → (*𝑟𝑓) = )
87fveq1d 6152 . . . . 5 (𝑓 = 𝑅 → ((*𝑟𝑓)‘𝑥) = ( 𝑥))
94, 8mpteq12dv 4698 . . . 4 (𝑓 = 𝑅 → (𝑥 ∈ (Base‘𝑓) ↦ ((*𝑟𝑓)‘𝑥)) = (𝑥𝐵 ↦ ( 𝑥)))
10 df-staf 18761 . . . 4 *rf = (𝑓 ∈ V ↦ (𝑥 ∈ (Base‘𝑓) ↦ ((*𝑟𝑓)‘𝑥)))
11 eqid 2626 . . . . . 6 (𝑥𝐵 ↦ ( 𝑥)) = (𝑥𝐵 ↦ ( 𝑥))
12 fvrn0 6174 . . . . . . 7 ( 𝑥) ∈ (ran ∪ {∅})
1312a1i 11 . . . . . 6 (𝑥𝐵 → ( 𝑥) ∈ (ran ∪ {∅}))
1411, 13fmpti 6340 . . . . 5 (𝑥𝐵 ↦ ( 𝑥)):𝐵⟶(ran ∪ {∅})
15 fvex 6160 . . . . . 6 (Base‘𝑅) ∈ V
163, 15eqeltri 2700 . . . . 5 𝐵 ∈ V
17 fvex 6160 . . . . . . . 8 (*𝑟𝑅) ∈ V
186, 17eqeltri 2700 . . . . . . 7 ∈ V
1918rnex 7048 . . . . . 6 ran ∈ V
20 p0ex 4818 . . . . . 6 {∅} ∈ V
2119, 20unex 6910 . . . . 5 (ran ∪ {∅}) ∈ V
22 fex2 7071 . . . . 5 (((𝑥𝐵 ↦ ( 𝑥)):𝐵⟶(ran ∪ {∅}) ∧ 𝐵 ∈ V ∧ (ran ∪ {∅}) ∈ V) → (𝑥𝐵 ↦ ( 𝑥)) ∈ V)
2314, 16, 21, 22mp3an 1421 . . . 4 (𝑥𝐵 ↦ ( 𝑥)) ∈ V
249, 10, 23fvmpt 6240 . . 3 (𝑅 ∈ V → (*rf𝑅) = (𝑥𝐵 ↦ ( 𝑥)))
25 fvprc 6144 . . . . 5 𝑅 ∈ V → (*rf𝑅) = ∅)
26 mpt0 5980 . . . . 5 (𝑥 ∈ ∅ ↦ ( 𝑥)) = ∅
2725, 26syl6eqr 2678 . . . 4 𝑅 ∈ V → (*rf𝑅) = (𝑥 ∈ ∅ ↦ ( 𝑥)))
28 fvprc 6144 . . . . . 6 𝑅 ∈ V → (Base‘𝑅) = ∅)
293, 28syl5eq 2672 . . . . 5 𝑅 ∈ V → 𝐵 = ∅)
3029mpteq1d 4703 . . . 4 𝑅 ∈ V → (𝑥𝐵 ↦ ( 𝑥)) = (𝑥 ∈ ∅ ↦ ( 𝑥)))
3127, 30eqtr4d 2663 . . 3 𝑅 ∈ V → (*rf𝑅) = (𝑥𝐵 ↦ ( 𝑥)))
3224, 31pm2.61i 176 . 2 (*rf𝑅) = (𝑥𝐵 ↦ ( 𝑥))
331, 32eqtri 2648 1 = (𝑥𝐵 ↦ ( 𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1480  wcel 1992  Vcvv 3191  cun 3558  c0 3896  {csn 4153  cmpt 4678  ran crn 5080  wf 5846  cfv 5850  Basecbs 15776  *𝑟cstv 15859  *rfcstf 18759
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fv 5858  df-staf 18761
This theorem is referenced by:  stafval  18764  staffn  18765  issrngd  18777
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