Theorem staffval 20944

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 6881	. . . . . 6 ⊢ (𝑓 = 𝑅 → (Base‘𝑓) = (Base‘𝑅))
3		staffval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
4	2, 3	eqtr4di 2816	. . . . 5 ⊢ (𝑓 = 𝑅 → (Base‘𝑓) = 𝐵)
5		fveq2 6881	. . . . . . 7 ⊢ (𝑓 = 𝑅 → (*𝑟‘𝑓) = (*𝑟‘𝑅))
6		staffval.i	. . . . . . 7 ⊢ ∗ = (*𝑟‘𝑅)
7	5, 6	eqtr4di 2816	. . . . . 6 ⊢ (𝑓 = 𝑅 → (*𝑟‘𝑓) = ∗ )
8	7	fveq1d 6883	. . . . 5 ⊢ (𝑓 = 𝑅 → ((*𝑟‘𝑓)‘𝑥) = ( ∗ ‘𝑥))
9	4, 8	mpteq12dv 5198	. . . 4 ⊢ (𝑓 = 𝑅 → (𝑥 ∈ (Base‘𝑓) ↦ ((*𝑟‘𝑓)‘𝑥)) = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)))
10		df-staf 20942	. . . 4 ⊢ *rf = (𝑓 ∈ V ↦ (𝑥 ∈ (Base‘𝑓) ↦ ((*𝑟‘𝑓)‘𝑥)))
11		eqid 2763	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)) = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥))
12		fvrn0 6909	. . . . . . 7 ⊢ ( ∗ ‘𝑥) ∈ (ran ∗ ∪ {∅})
13	12	a1i 11	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 → ( ∗ ‘𝑥) ∈ (ran ∗ ∪ {∅}))
14	11, 13	fmpti 7107	. . . . 5 ⊢ (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)):𝐵⟶(ran ∗ ∪ {∅})
15	3	fvexi 6895	. . . . 5 ⊢ 𝐵 ∈ V
16	6	fvexi 6895	. . . . . . 7 ⊢ ∗ ∈ V
17	16	rnex 7903	. . . . . 6 ⊢ ran ∗ ∈ V
18		p0ex 5355	. . . . . 6 ⊢ {∅} ∈ V
19	17, 18	unex 7742	. . . . 5 ⊢ (ran ∗ ∪ {∅}) ∈ V
20		fex2 7929	. . . . 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 6989	. . 3 ⊢ (𝑅 ∈ V → (*rf‘𝑅) = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)))
23		fvprc 6873	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (*rf‘𝑅) = ∅)
24		mpt0 6677	. . . . 5 ⊢ (𝑥 ∈ ∅ ↦ ( ∗ ‘𝑥)) = ∅
25	23, 24	eqtr4di 2816	. . . 4 ⊢ (¬ 𝑅 ∈ V → (*rf‘𝑅) = (𝑥 ∈ ∅ ↦ ( ∗ ‘𝑥)))
26		fvprc 6873	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → (Base‘𝑅) = ∅)
27	3, 26	eqtrid 2810	. . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝐵 = ∅)
28	27	mpteq1d 5201	. . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)) = (𝑥 ∈ ∅ ↦ ( ∗ ‘𝑥)))
29	25, 28	eqtr4d 2801	. . 3 ⊢ (¬ 𝑅 ∈ V → (*rf‘𝑅) = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)))
30	22, 29	pm2.61i 184	. 2 ⊢ (*rf‘𝑅) = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥))
31	1, 30	eqtri 2786	1 ⊢ ∙ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥))

Syntax hints:  ¬ wn 3   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ∪ cun 3903  ∅c0 4286  {csn 4589   ↦ cmpt 5192  ran crn 5662  ⟶wf 6532  ‘cfv 6536  Basecbs 17264  *_𝑟cstv 17307  *_rfcstf 20940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-staf 20942

This theorem is referenced by:  stafval  20945  staffn  20946  issrngd  20958