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Theorem staffval 19069
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 6353 . . . . . 6 (𝑓 = 𝑅 → (Base‘𝑓) = (Base‘𝑅))
3 staffval.b . . . . . 6 𝐵 = (Base‘𝑅)
42, 3syl6eqr 2812 . . . . 5 (𝑓 = 𝑅 → (Base‘𝑓) = 𝐵)
5 fveq2 6353 . . . . . . 7 (𝑓 = 𝑅 → (*𝑟𝑓) = (*𝑟𝑅))
6 staffval.i . . . . . . 7 = (*𝑟𝑅)
75, 6syl6eqr 2812 . . . . . 6 (𝑓 = 𝑅 → (*𝑟𝑓) = )
87fveq1d 6355 . . . . 5 (𝑓 = 𝑅 → ((*𝑟𝑓)‘𝑥) = ( 𝑥))
94, 8mpteq12dv 4885 . . . 4 (𝑓 = 𝑅 → (𝑥 ∈ (Base‘𝑓) ↦ ((*𝑟𝑓)‘𝑥)) = (𝑥𝐵 ↦ ( 𝑥)))
10 df-staf 19067 . . . 4 *rf = (𝑓 ∈ V ↦ (𝑥 ∈ (Base‘𝑓) ↦ ((*𝑟𝑓)‘𝑥)))
11 eqid 2760 . . . . . 6 (𝑥𝐵 ↦ ( 𝑥)) = (𝑥𝐵 ↦ ( 𝑥))
12 fvrn0 6378 . . . . . . 7 ( 𝑥) ∈ (ran ∪ {∅})
1312a1i 11 . . . . . 6 (𝑥𝐵 → ( 𝑥) ∈ (ran ∪ {∅}))
1411, 13fmpti 6547 . . . . 5 (𝑥𝐵 ↦ ( 𝑥)):𝐵⟶(ran ∪ {∅})
15 fvex 6363 . . . . . 6 (Base‘𝑅) ∈ V
163, 15eqeltri 2835 . . . . 5 𝐵 ∈ V
17 fvex 6363 . . . . . . . 8 (*𝑟𝑅) ∈ V
186, 17eqeltri 2835 . . . . . . 7 ∈ V
1918rnex 7266 . . . . . 6 ran ∈ V
20 p0ex 5002 . . . . . 6 {∅} ∈ V
2119, 20unex 7122 . . . . 5 (ran ∪ {∅}) ∈ V
22 fex2 7287 . . . . 5 (((𝑥𝐵 ↦ ( 𝑥)):𝐵⟶(ran ∪ {∅}) ∧ 𝐵 ∈ V ∧ (ran ∪ {∅}) ∈ V) → (𝑥𝐵 ↦ ( 𝑥)) ∈ V)
2314, 16, 21, 22mp3an 1573 . . . 4 (𝑥𝐵 ↦ ( 𝑥)) ∈ V
249, 10, 23fvmpt 6445 . . 3 (𝑅 ∈ V → (*rf𝑅) = (𝑥𝐵 ↦ ( 𝑥)))
25 fvprc 6347 . . . . 5 𝑅 ∈ V → (*rf𝑅) = ∅)
26 mpt0 6182 . . . . 5 (𝑥 ∈ ∅ ↦ ( 𝑥)) = ∅
2725, 26syl6eqr 2812 . . . 4 𝑅 ∈ V → (*rf𝑅) = (𝑥 ∈ ∅ ↦ ( 𝑥)))
28 fvprc 6347 . . . . . 6 𝑅 ∈ V → (Base‘𝑅) = ∅)
293, 28syl5eq 2806 . . . . 5 𝑅 ∈ V → 𝐵 = ∅)
3029mpteq1d 4890 . . . 4 𝑅 ∈ V → (𝑥𝐵 ↦ ( 𝑥)) = (𝑥 ∈ ∅ ↦ ( 𝑥)))
3127, 30eqtr4d 2797 . . 3 𝑅 ∈ V → (*rf𝑅) = (𝑥𝐵 ↦ ( 𝑥)))
3224, 31pm2.61i 176 . 2 (*rf𝑅) = (𝑥𝐵 ↦ ( 𝑥))
331, 32eqtri 2782 1 = (𝑥𝐵 ↦ ( 𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1632  wcel 2139  Vcvv 3340  cun 3713  c0 4058  {csn 4321  cmpt 4881  ran crn 5267  wf 6045  cfv 6049  Basecbs 16079  *𝑟cstv 16165  *rfcstf 19065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-staf 19067
This theorem is referenced by:  stafval  19070  staffn  19071  issrngd  19083
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