Theorem staffval 19620

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 6672	. . . . . 6 ⊢ (𝑓 = 𝑅 → (Base‘𝑓) = (Base‘𝑅))
3		staffval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
4	2, 3	syl6eqr 2876	. . . . 5 ⊢ (𝑓 = 𝑅 → (Base‘𝑓) = 𝐵)
5		fveq2 6672	. . . . . . 7 ⊢ (𝑓 = 𝑅 → (*𝑟‘𝑓) = (*𝑟‘𝑅))
6		staffval.i	. . . . . . 7 ⊢ ∗ = (*𝑟‘𝑅)
7	5, 6	syl6eqr 2876	. . . . . 6 ⊢ (𝑓 = 𝑅 → (*𝑟‘𝑓) = ∗ )
8	7	fveq1d 6674	. . . . 5 ⊢ (𝑓 = 𝑅 → ((*𝑟‘𝑓)‘𝑥) = ( ∗ ‘𝑥))
9	4, 8	mpteq12dv 5153	. . . 4 ⊢ (𝑓 = 𝑅 → (𝑥 ∈ (Base‘𝑓) ↦ ((*𝑟‘𝑓)‘𝑥)) = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)))
10		df-staf 19618	. . . 4 ⊢ *rf = (𝑓 ∈ V ↦ (𝑥 ∈ (Base‘𝑓) ↦ ((*𝑟‘𝑓)‘𝑥)))
11		eqid 2823	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)) = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥))
12		fvrn0 6700	. . . . . . 7 ⊢ ( ∗ ‘𝑥) ∈ (ran ∗ ∪ {∅})
13	12	a1i 11	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 → ( ∗ ‘𝑥) ∈ (ran ∗ ∪ {∅}))
14	11, 13	fmpti 6878	. . . . 5 ⊢ (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)):𝐵⟶(ran ∗ ∪ {∅})
15	3	fvexi 6686	. . . . 5 ⊢ 𝐵 ∈ V
16	6	fvexi 6686	. . . . . . 7 ⊢ ∗ ∈ V
17	16	rnex 7619	. . . . . 6 ⊢ ran ∗ ∈ V
18		p0ex 5287	. . . . . 6 ⊢ {∅} ∈ V
19	17, 18	unex 7471	. . . . 5 ⊢ (ran ∗ ∪ {∅}) ∈ V
20		fex2 7640	. . . . 5 ⊢ (((𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)):𝐵⟶(ran ∗ ∪ {∅}) ∧ 𝐵 ∈ V ∧ (ran ∗ ∪ {∅}) ∈ V) → (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)) ∈ V)
21	14, 15, 19, 20	mp3an 1457	. . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)) ∈ V
22	9, 10, 21	fvmpt 6770	. . 3 ⊢ (𝑅 ∈ V → (*rf‘𝑅) = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)))
23		fvprc 6665	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (*rf‘𝑅) = ∅)
24		mpt0 6492	. . . . 5 ⊢ (𝑥 ∈ ∅ ↦ ( ∗ ‘𝑥)) = ∅
25	23, 24	syl6eqr 2876	. . . 4 ⊢ (¬ 𝑅 ∈ V → (*rf‘𝑅) = (𝑥 ∈ ∅ ↦ ( ∗ ‘𝑥)))
26		fvprc 6665	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → (Base‘𝑅) = ∅)
27	3, 26	syl5eq 2870	. . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝐵 = ∅)
28	27	mpteq1d 5157	. . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)) = (𝑥 ∈ ∅ ↦ ( ∗ ‘𝑥)))
29	25, 28	eqtr4d 2861	. . 3 ⊢ (¬ 𝑅 ∈ V → (*rf‘𝑅) = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)))
30	22, 29	pm2.61i 184	. 2 ⊢ (*rf‘𝑅) = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥))
31	1, 30	eqtri 2846	1 ⊢ ∙ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥))

Syntax hints:  ¬ wn 3   = wceq 1537   ∈ wcel 2114  Vcvv 3496   ∪ cun 3936  ∅c0 4293  {csn 4569   ↦ cmpt 5148  ran crn 5558  ⟶wf 6353  ‘cfv 6357  Basecbs 16485  *_𝑟cstv 16569  *_rfcstf 19616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fv 6365  df-staf 19618

This theorem is referenced by:  stafval  19621  staffn  19622  issrngd  19634