Theorem staffval 20088

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	⊢ ∗ = (*𝑟‘𝑅)
staffval.f	⊢ ∙ = (*rf‘𝑅)

Assertion

Ref	Expression
staffval	⊢ ∙ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘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.

Step	Hyp	Ref	Expression
1		staffval.f	. 2 ⊢ ∙ = (*rf‘𝑅)
2		fveq2 6768	. . . . . 6 ⊢ (𝑓 = 𝑅 → (Base‘𝑓) = (Base‘𝑅))
3		staffval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
4	2, 3	eqtr4di 2797	. . . . 5 ⊢ (𝑓 = 𝑅 → (Base‘𝑓) = 𝐵)
5		fveq2 6768	. . . . . . 7 ⊢ (𝑓 = 𝑅 → (*𝑟‘𝑓) = (*𝑟‘𝑅))
6		staffval.i	. . . . . . 7 ⊢ ∗ = (*𝑟‘𝑅)
7	5, 6	eqtr4di 2797	. . . . . 6 ⊢ (𝑓 = 𝑅 → (*𝑟‘𝑓) = ∗ )
8	7	fveq1d 6770	. . . . 5 ⊢ (𝑓 = 𝑅 → ((*𝑟‘𝑓)‘𝑥) = ( ∗ ‘𝑥))
9	4, 8	mpteq12dv 5169	. . . 4 ⊢ (𝑓 = 𝑅 → (𝑥 ∈ (Base‘𝑓) ↦ ((*𝑟‘𝑓)‘𝑥)) = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)))
10		df-staf 20086	. . . 4 ⊢ *rf = (𝑓 ∈ V ↦ (𝑥 ∈ (Base‘𝑓) ↦ ((*𝑟‘𝑓)‘𝑥)))
11		eqid 2739	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)) = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥))
12		fvrn0 6796	. . . . . . 7 ⊢ ( ∗ ‘𝑥) ∈ (ran ∗ ∪ {∅})
13	12	a1i 11	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 → ( ∗ ‘𝑥) ∈ (ran ∗ ∪ {∅}))
14	11, 13	fmpti 6980	. . . . 5 ⊢ (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)):𝐵⟶(ran ∗ ∪ {∅})
15	3	fvexi 6782	. . . . 5 ⊢ 𝐵 ∈ V
16	6	fvexi 6782	. . . . . . 7 ⊢ ∗ ∈ V
17	16	rnex 7746	. . . . . 6 ⊢ ran ∗ ∈ V
18		p0ex 5310	. . . . . 6 ⊢ {∅} ∈ V
19	17, 18	unex 7587	. . . . 5 ⊢ (ran ∗ ∪ {∅}) ∈ V
20		fex2 7767	. . . . 5 ⊢ (((𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)):𝐵⟶(ran ∗ ∪ {∅}) ∧ 𝐵 ∈ V ∧ (ran ∗ ∪ {∅}) ∈ V) → (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)) ∈ V)
21	14, 15, 19, 20	mp3an 1459	. . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)) ∈ V
22	9, 10, 21	fvmpt 6869	. . 3 ⊢ (𝑅 ∈ V → (*rf‘𝑅) = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)))
23		fvprc 6760	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (*rf‘𝑅) = ∅)
24		mpt0 6571	. . . . 5 ⊢ (𝑥 ∈ ∅ ↦ ( ∗ ‘𝑥)) = ∅
25	23, 24	eqtr4di 2797	. . . 4 ⊢ (¬ 𝑅 ∈ V → (*rf‘𝑅) = (𝑥 ∈ ∅ ↦ ( ∗ ‘𝑥)))
26		fvprc 6760	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → (Base‘𝑅) = ∅)
27	3, 26	eqtrid 2791	. . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝐵 = ∅)
28	27	mpteq1d 5173	. . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)) = (𝑥 ∈ ∅ ↦ ( ∗ ‘𝑥)))
29	25, 28	eqtr4d 2782	. . 3 ⊢ (¬ 𝑅 ∈ V → (*rf‘𝑅) = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)))
30	22, 29	pm2.61i 182	. 2 ⊢ (*rf‘𝑅) = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥))
31	1, 30	eqtri 2767	1 ⊢ ∙ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥))

Syntax hints:  ¬ wn 3   = wceq 1541   ∈ wcel 2109  Vcvv 3430   ∪ cun 3889  ∅c0 4261  {csn 4566   ↦ cmpt 5161  ran crn 5589  ⟶wf 6426  ‘cfv 6430  Basecbs 16893  *_𝑟cstv 16945  *_rfcstf 20084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-fv 6438  df-staf 20086

This theorem is referenced by:  stafval  20089  staffn  20090  issrngd  20102