Theorem fsplit 7282

Description: A function that can be used to feed a common value to both operands of an operation. Use as the second argument of a composition with the function of fpar 7281 in order to build compound functions such as 𝑦 = ((√‘𝑥) + (abs‘𝑥)). (Contributed by NM, 17-Sep-2007.)

Assertion

Ref	Expression
fsplit	⊢ ◡(1st ↾ I ) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)

Proof of Theorem fsplit

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. . . . 5 ⊢ 𝑥 ∈ V
2		vex 3203	. . . . 5 ⊢ 𝑦 ∈ V
3	1, 2	brcnv 5305	. . . 4 ⊢ (𝑥◡(1st ↾ I )𝑦 ↔ 𝑦(1st ↾ I )𝑥)
4	1	brres 5402	. . . . 5 ⊢ (𝑦(1st ↾ I )𝑥 ↔ (𝑦1st 𝑥 ∧ 𝑦 ∈ I ))
5		19.42v 1918	. . . . . . 7 ⊢ (∃𝑧((1st ‘𝑦) = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩) ↔ ((1st ‘𝑦) = 𝑥 ∧ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩))
6		vex 3203	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
7	6, 6	op1std 7178	. . . . . . . . . 10 ⊢ (𝑦 = ⟨𝑧, 𝑧⟩ → (1st ‘𝑦) = 𝑧)
8	7	eqeq1d 2624	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑧, 𝑧⟩ → ((1st ‘𝑦) = 𝑥 ↔ 𝑧 = 𝑥))
9	8	pm5.32ri 670	. . . . . . . 8 ⊢ (((1st ‘𝑦) = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩) ↔ (𝑧 = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩))
10	9	exbii 1774	. . . . . . 7 ⊢ (∃𝑧((1st ‘𝑦) = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩) ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩))
11		fo1st 7188	. . . . . . . . . 10 ⊢ 1st :V–onto→V
12		fofn 6117	. . . . . . . . . 10 ⊢ (1st :V–onto→V → 1st Fn V)
13	11, 12	ax-mp 5	. . . . . . . . 9 ⊢ 1st Fn V
14		fnbrfvb 6236	. . . . . . . . 9 ⊢ ((1st Fn V ∧ 𝑦 ∈ V) → ((1st ‘𝑦) = 𝑥 ↔ 𝑦1st 𝑥))
15	13, 2, 14	mp2an 708	. . . . . . . 8 ⊢ ((1st ‘𝑦) = 𝑥 ↔ 𝑦1st 𝑥)
16		dfid2 5027	. . . . . . . . . 10 ⊢ I = {⟨𝑧, 𝑧⟩ ∣ 𝑧 = 𝑧}
17	16	eleq2i 2693	. . . . . . . . 9 ⊢ (𝑦 ∈ I ↔ 𝑦 ∈ {⟨𝑧, 𝑧⟩ ∣ 𝑧 = 𝑧})
18		nfe1 2027	. . . . . . . . . . 11 ⊢ Ⅎ𝑧∃𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧)
19	18	19.9 2072	. . . . . . . . . 10 ⊢ (∃𝑧∃𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧) ↔ ∃𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧))
20		elopab 4983	. . . . . . . . . 10 ⊢ (𝑦 ∈ {⟨𝑧, 𝑧⟩ ∣ 𝑧 = 𝑧} ↔ ∃𝑧∃𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧))
21		equid 1939	. . . . . . . . . . . 12 ⊢ 𝑧 = 𝑧
22	21	biantru 526	. . . . . . . . . . 11 ⊢ (𝑦 = ⟨𝑧, 𝑧⟩ ↔ (𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧))
23	22	exbii 1774	. . . . . . . . . 10 ⊢ (∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩ ↔ ∃𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧))
24	19, 20, 23	3bitr4i 292	. . . . . . . . 9 ⊢ (𝑦 ∈ {⟨𝑧, 𝑧⟩ ∣ 𝑧 = 𝑧} ↔ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩)
25	17, 24	bitr2i 265	. . . . . . . 8 ⊢ (∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩ ↔ 𝑦 ∈ I )
26	15, 25	anbi12i 733	. . . . . . 7 ⊢ (((1st ‘𝑦) = 𝑥 ∧ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩) ↔ (𝑦1st 𝑥 ∧ 𝑦 ∈ I ))
27	5, 10, 26	3bitr3ri 291	. . . . . 6 ⊢ ((𝑦1st 𝑥 ∧ 𝑦 ∈ I ) ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩))
28		id 22	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → 𝑧 = 𝑥)
29	28, 28	opeq12d 4410	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ⟨𝑧, 𝑧⟩ = ⟨𝑥, 𝑥⟩)
30	29	eqeq2d 2632	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑦 = ⟨𝑧, 𝑧⟩ ↔ 𝑦 = ⟨𝑥, 𝑥⟩))
31	1, 30	ceqsexv 3242	. . . . . 6 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩) ↔ 𝑦 = ⟨𝑥, 𝑥⟩)
32	27, 31	bitri 264	. . . . 5 ⊢ ((𝑦1st 𝑥 ∧ 𝑦 ∈ I ) ↔ 𝑦 = ⟨𝑥, 𝑥⟩)
33	4, 32	bitri 264	. . . 4 ⊢ (𝑦(1st ↾ I )𝑥 ↔ 𝑦 = ⟨𝑥, 𝑥⟩)
34	3, 33	bitri 264	. . 3 ⊢ (𝑥◡(1st ↾ I )𝑦 ↔ 𝑦 = ⟨𝑥, 𝑥⟩)
35	34	opabbii 4717	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥◡(1st ↾ I )𝑦} = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = ⟨𝑥, 𝑥⟩}
36		relcnv 5503	. . 3 ⊢ Rel ◡(1st ↾ I )
37		dfrel4v 5584	. . 3 ⊢ (Rel ◡(1st ↾ I ) ↔ ◡(1st ↾ I ) = {⟨𝑥, 𝑦⟩ ∣ 𝑥◡(1st ↾ I )𝑦})
38	36, 37	mpbi 220	. 2 ⊢ ◡(1st ↾ I ) = {⟨𝑥, 𝑦⟩ ∣ 𝑥◡(1st ↾ I )𝑦}
39		mptv 4751	. 2 ⊢ (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = ⟨𝑥, 𝑥⟩}
40	35, 38, 39	3eqtr4i 2654	1 ⊢ ◡(1st ↾ I ) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)

Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  Vcvv 3200  ⟨cop 4183   class class class wbr 4653  {copab 4712   ↦ cmpt 4729   I cid 5023  ^◡ccnv 5113   ↾ cres 5116  Rel wrel 5119   Fn wfn 5883  –onto→wfo 5886  ‘cfv 5888  1^st c1st 7166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-1st 7168

This theorem is referenced by: (None)