Theorem staffval 21107

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 6890	. . . . . 6 ⊢ (𝑓 = 𝑅 → (Base‘𝑓) = (Base‘𝑅))
3		staffval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
4	2, 3	eqtr4di 2813	. . . . 5 ⊢ (𝑓 = 𝑅 → (Base‘𝑓) = 𝐵)
5		fveq2 6890	. . . . . . 7 ⊢ (𝑓 = 𝑅 → (*𝑟‘𝑓) = (*𝑟‘𝑅))
6		staffval.i	. . . . . . 7 ⊢ ∗ = (*𝑟‘𝑅)
7	5, 6	eqtr4di 2813	. . . . . 6 ⊢ (𝑓 = 𝑅 → (*𝑟‘𝑓) = ∗ )
8	7	fveq1d 6892	. . . . 5 ⊢ (𝑓 = 𝑅 → ((*𝑟‘𝑓)‘𝑥) = ( ∗ ‘𝑥))
9	4, 8	mpteq12dv 5191	. . . 4 ⊢ (𝑓 = 𝑅 → (𝑥 ∈ (Base‘𝑓) ↦ ((*𝑟‘𝑓)‘𝑥)) = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)))
10		df-staf 21105	. . . 4 ⊢ *rf = (𝑓 ∈ V ↦ (𝑥 ∈ (Base‘𝑓) ↦ ((*𝑟‘𝑓)‘𝑥)))
11		eqid 2760	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)) = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥))
12		fvrn0 6918	. . . . . . 7 ⊢ ( ∗ ‘𝑥) ∈ (ran ∗ ∪ {∅})
13	12	a1i 11	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 → ( ∗ ‘𝑥) ∈ (ran ∗ ∪ {∅}))
14	11, 13	fmpti 7117	. . . . 5 ⊢ (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)):𝐵⟶(ran ∗ ∪ {∅})
15	3	fvexi 6904	. . . . 5 ⊢ 𝐵 ∈ V
16	6	fvexi 6904	. . . . . . 7 ⊢ ∗ ∈ V
17	16	rnex 7926	. . . . . 6 ⊢ ran ∗ ∈ V
18		p0ex 5346	. . . . . 6 ⊢ {∅} ∈ V
19	17, 18	unex 7765	. . . . 5 ⊢ (ran ∗ ∪ {∅}) ∈ V
20		fex2 7952	. . . . 5 ⊢ (((𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)):𝐵⟶(ran ∗ ∪ {∅}) ∧ 𝐵 ∈ V ∧ (ran ∗ ∪ {∅}) ∈ V) → (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)) ∈ V)
21	14, 15, 19, 20	mp3an 1490	. . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)) ∈ V
22	9, 10, 21	fvmpt 6998	. . 3 ⊢ (𝑅 ∈ V → (*rf‘𝑅) = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)))
23		fvprc 6882	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (*rf‘𝑅) = ∅)
24		mpt0 6686	. . . . 5 ⊢ (𝑥 ∈ ∅ ↦ ( ∗ ‘𝑥)) = ∅
25	23, 24	eqtr4di 2813	. . . 4 ⊢ (¬ 𝑅 ∈ V → (*rf‘𝑅) = (𝑥 ∈ ∅ ↦ ( ∗ ‘𝑥)))
26		fvprc 6882	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → (Base‘𝑅) = ∅)
27	3, 26	eqtrid 2807	. . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝐵 = ∅)
28	27	mpteq1d 5194	. . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)) = (𝑥 ∈ ∅ ↦ ( ∗ ‘𝑥)))
29	25, 28	eqtr4d 2798	. . 3 ⊢ (¬ 𝑅 ∈ V → (*rf‘𝑅) = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)))
30	22, 29	pm2.61i 184	. 2 ⊢ (*rf‘𝑅) = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥))
31	1, 30	eqtri 2783	1 ⊢ ∙ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥))

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1569   ∈ wcel 2144  Vcvv 3450   ∪ cun 4148  ∅c0 4277  {csn 4584   ↦ cmpt 5185  ran crn 5652  ⟶wf 6539  ‘cfv 6543  Basecbs 17396  *_𝑟cstv 17439  *_rfcstf 21103

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-staf 21105

This theorem is used by:  stafval  21108  staffn  21109  issrngd  21121