Theorem staffval 19740

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 6677	. . . . . 6 ⊢ (𝑓 = 𝑅 → (Base‘𝑓) = (Base‘𝑅))
3		staffval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
4	2, 3	eqtr4di 2792	. . . . 5 ⊢ (𝑓 = 𝑅 → (Base‘𝑓) = 𝐵)
5		fveq2 6677	. . . . . . 7 ⊢ (𝑓 = 𝑅 → (*𝑟‘𝑓) = (*𝑟‘𝑅))
6		staffval.i	. . . . . . 7 ⊢ ∗ = (*𝑟‘𝑅)
7	5, 6	eqtr4di 2792	. . . . . 6 ⊢ (𝑓 = 𝑅 → (*𝑟‘𝑓) = ∗ )
8	7	fveq1d 6679	. . . . 5 ⊢ (𝑓 = 𝑅 → ((*𝑟‘𝑓)‘𝑥) = ( ∗ ‘𝑥))
9	4, 8	mpteq12dv 5116	. . . 4 ⊢ (𝑓 = 𝑅 → (𝑥 ∈ (Base‘𝑓) ↦ ((*𝑟‘𝑓)‘𝑥)) = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)))
10		df-staf 19738	. . . 4 ⊢ *rf = (𝑓 ∈ V ↦ (𝑥 ∈ (Base‘𝑓) ↦ ((*𝑟‘𝑓)‘𝑥)))
11		eqid 2739	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)) = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥))
12		fvrn0 6705	. . . . . . 7 ⊢ ( ∗ ‘𝑥) ∈ (ran ∗ ∪ {∅})
13	12	a1i 11	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 → ( ∗ ‘𝑥) ∈ (ran ∗ ∪ {∅}))
14	11, 13	fmpti 6889	. . . . 5 ⊢ (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)):𝐵⟶(ran ∗ ∪ {∅})
15	3	fvexi 6691	. . . . 5 ⊢ 𝐵 ∈ V
16	6	fvexi 6691	. . . . . . 7 ⊢ ∗ ∈ V
17	16	rnex 7646	. . . . . 6 ⊢ ran ∗ ∈ V
18		p0ex 5252	. . . . . 6 ⊢ {∅} ∈ V
19	17, 18	unex 7490	. . . . 5 ⊢ (ran ∗ ∪ {∅}) ∈ V
20		fex2 7667	. . . . 5 ⊢ (((𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)):𝐵⟶(ran ∗ ∪ {∅}) ∧ 𝐵 ∈ V ∧ (ran ∗ ∪ {∅}) ∈ V) → (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)) ∈ V)
21	14, 15, 19, 20	mp3an 1462	. . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)) ∈ V
22	9, 10, 21	fvmpt 6778	. . 3 ⊢ (𝑅 ∈ V → (*rf‘𝑅) = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)))
23		fvprc 6669	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (*rf‘𝑅) = ∅)
24		mpt0 6480	. . . . 5 ⊢ (𝑥 ∈ ∅ ↦ ( ∗ ‘𝑥)) = ∅
25	23, 24	eqtr4di 2792	. . . 4 ⊢ (¬ 𝑅 ∈ V → (*rf‘𝑅) = (𝑥 ∈ ∅ ↦ ( ∗ ‘𝑥)))
26		fvprc 6669	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → (Base‘𝑅) = ∅)
27	3, 26	syl5eq 2786	. . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝐵 = ∅)
28	27	mpteq1d 5120	. . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)) = (𝑥 ∈ ∅ ↦ ( ∗ ‘𝑥)))
29	25, 28	eqtr4d 2777	. . 3 ⊢ (¬ 𝑅 ∈ V → (*rf‘𝑅) = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)))
30	22, 29	pm2.61i 185	. 2 ⊢ (*rf‘𝑅) = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥))
31	1, 30	eqtri 2762	1 ⊢ ∙ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥))

Syntax hints:  ¬ wn 3   = wceq 1542   ∈ wcel 2114  Vcvv 3399   ∪ cun 3842  ∅c0 4212  {csn 4517   ↦ cmpt 5111  ran crn 5527  ⟶wf 6336  ‘cfv 6340  Basecbs 16589  *_𝑟cstv 16673  *_rfcstf 19736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297  ax-un 7482

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3401  df-sbc 3682  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-fv 6348  df-staf 19738

This theorem is referenced by:  stafval  19741  staffn  19742  issrngd  19754