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Theorem staffval 19057
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 6332 . . . . . 6 (𝑓 = 𝑅 → (Base‘𝑓) = (Base‘𝑅))
3 staffval.b . . . . . 6 𝐵 = (Base‘𝑅)
42, 3syl6eqr 2823 . . . . 5 (𝑓 = 𝑅 → (Base‘𝑓) = 𝐵)
5 fveq2 6332 . . . . . . 7 (𝑓 = 𝑅 → (*𝑟𝑓) = (*𝑟𝑅))
6 staffval.i . . . . . . 7 = (*𝑟𝑅)
75, 6syl6eqr 2823 . . . . . 6 (𝑓 = 𝑅 → (*𝑟𝑓) = )
87fveq1d 6334 . . . . 5 (𝑓 = 𝑅 → ((*𝑟𝑓)‘𝑥) = ( 𝑥))
94, 8mpteq12dv 4867 . . . 4 (𝑓 = 𝑅 → (𝑥 ∈ (Base‘𝑓) ↦ ((*𝑟𝑓)‘𝑥)) = (𝑥𝐵 ↦ ( 𝑥)))
10 df-staf 19055 . . . 4 *rf = (𝑓 ∈ V ↦ (𝑥 ∈ (Base‘𝑓) ↦ ((*𝑟𝑓)‘𝑥)))
11 eqid 2771 . . . . . 6 (𝑥𝐵 ↦ ( 𝑥)) = (𝑥𝐵 ↦ ( 𝑥))
12 fvrn0 6357 . . . . . . 7 ( 𝑥) ∈ (ran ∪ {∅})
1312a1i 11 . . . . . 6 (𝑥𝐵 → ( 𝑥) ∈ (ran ∪ {∅}))
1411, 13fmpti 6525 . . . . 5 (𝑥𝐵 ↦ ( 𝑥)):𝐵⟶(ran ∪ {∅})
15 fvex 6342 . . . . . 6 (Base‘𝑅) ∈ V
163, 15eqeltri 2846 . . . . 5 𝐵 ∈ V
17 fvex 6342 . . . . . . . 8 (*𝑟𝑅) ∈ V
186, 17eqeltri 2846 . . . . . . 7 ∈ V
1918rnex 7247 . . . . . 6 ran ∈ V
20 p0ex 4984 . . . . . 6 {∅} ∈ V
2119, 20unex 7103 . . . . 5 (ran ∪ {∅}) ∈ V
22 fex2 7268 . . . . 5 (((𝑥𝐵 ↦ ( 𝑥)):𝐵⟶(ran ∪ {∅}) ∧ 𝐵 ∈ V ∧ (ran ∪ {∅}) ∈ V) → (𝑥𝐵 ↦ ( 𝑥)) ∈ V)
2314, 16, 21, 22mp3an 1572 . . . 4 (𝑥𝐵 ↦ ( 𝑥)) ∈ V
249, 10, 23fvmpt 6424 . . 3 (𝑅 ∈ V → (*rf𝑅) = (𝑥𝐵 ↦ ( 𝑥)))
25 fvprc 6326 . . . . 5 𝑅 ∈ V → (*rf𝑅) = ∅)
26 mpt0 6161 . . . . 5 (𝑥 ∈ ∅ ↦ ( 𝑥)) = ∅
2725, 26syl6eqr 2823 . . . 4 𝑅 ∈ V → (*rf𝑅) = (𝑥 ∈ ∅ ↦ ( 𝑥)))
28 fvprc 6326 . . . . . 6 𝑅 ∈ V → (Base‘𝑅) = ∅)
293, 28syl5eq 2817 . . . . 5 𝑅 ∈ V → 𝐵 = ∅)
3029mpteq1d 4872 . . . 4 𝑅 ∈ V → (𝑥𝐵 ↦ ( 𝑥)) = (𝑥 ∈ ∅ ↦ ( 𝑥)))
3127, 30eqtr4d 2808 . . 3 𝑅 ∈ V → (*rf𝑅) = (𝑥𝐵 ↦ ( 𝑥)))
3224, 31pm2.61i 176 . 2 (*rf𝑅) = (𝑥𝐵 ↦ ( 𝑥))
331, 32eqtri 2793 1 = (𝑥𝐵 ↦ ( 𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1631  wcel 2145  Vcvv 3351  cun 3721  c0 4063  {csn 4316  cmpt 4863  ran crn 5250  wf 6027  cfv 6031  Basecbs 16064  *𝑟cstv 16151  *rfcstf 19053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-staf 19055
This theorem is referenced by:  stafval  19058  staffn  19059  issrngd  19071
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