Theorem fsplit 8113

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 8112 in order to build compound functions such as (𝑥 ∈ (0[,)+∞) ↦ ((√‘𝑥) + (sin‘𝑥))). (Contributed by NM, 17-Sep-2007.) Replace use of dfid2 5560 with df-id 5558. (Revised by BJ, 31-Dec-2023.)

Assertion

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

Proof of Theorem fsplit

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

Step	Hyp	Ref	Expression
1		vex 3459	. . . . 5 ⊢ 𝑥 ∈ V
2		vex 3459	. . . . 5 ⊢ 𝑦 ∈ V
3	1, 2	brcnv 5870	. . . 4 ⊢ (𝑥◡(1st ↾ I )𝑦 ↔ 𝑦(1st ↾ I )𝑥)
4	1	brresi 5989	. . . 4 ⊢ (𝑦(1st ↾ I )𝑥 ↔ (𝑦 ∈ I ∧ 𝑦1st 𝑥))
5		19.42v 1983	. . . . . 6 ⊢ (∃𝑧((1st ‘𝑦) = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩) ↔ ((1st ‘𝑦) = 𝑥 ∧ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩))
6		vex 3459	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
7	6, 6	op1std 7997	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑧, 𝑧⟩ → (1st ‘𝑦) = 𝑧)
8	7	eqeq1d 2765	. . . . . . . 8 ⊢ (𝑦 = ⟨𝑧, 𝑧⟩ → ((1st ‘𝑦) = 𝑥 ↔ 𝑧 = 𝑥))
9	8	pm5.32ri 585	. . . . . . 7 ⊢ (((1st ‘𝑦) = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩) ↔ (𝑧 = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩))
10	9	exbii 1878	. . . . . 6 ⊢ (∃𝑧((1st ‘𝑦) = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩) ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩))
11		fo1st 8007	. . . . . . . . 9 ⊢ 1st :V–onto→V
12		fofn 6796	. . . . . . . . 9 ⊢ (1st :V–onto→V → 1st Fn V)
13	11, 12	ax-mp 5	. . . . . . . 8 ⊢ 1st Fn V
14		fnbrfvb 6933	. . . . . . . 8 ⊢ ((1st Fn V ∧ 𝑦 ∈ V) → ((1st ‘𝑦) = 𝑥 ↔ 𝑦1st 𝑥))
15	13, 2, 14	mp2an 704	. . . . . . 7 ⊢ ((1st ‘𝑦) = 𝑥 ↔ 𝑦1st 𝑥)
16		df-id 5558	. . . . . . . . 9 ⊢ I = {⟨𝑧, 𝑡⟩ ∣ 𝑧 = 𝑡}
17	16	eleq2i 2855	. . . . . . . 8 ⊢ (𝑦 ∈ I ↔ 𝑦 ∈ {⟨𝑧, 𝑡⟩ ∣ 𝑧 = 𝑡})
18		elopab 5513	. . . . . . . 8 ⊢ (𝑦 ∈ {⟨𝑧, 𝑡⟩ ∣ 𝑧 = 𝑡} ↔ ∃𝑧∃𝑡(𝑦 = ⟨𝑧, 𝑡⟩ ∧ 𝑧 = 𝑡))
19		ancom 465	. . . . . . . . . . . 12 ⊢ ((𝑦 = ⟨𝑧, 𝑡⟩ ∧ 𝑧 = 𝑡) ↔ (𝑧 = 𝑡 ∧ 𝑦 = ⟨𝑧, 𝑡⟩))
20		equcom 2048	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑡 ↔ 𝑡 = 𝑧)
21	20	anbi1i 635	. . . . . . . . . . . 12 ⊢ ((𝑧 = 𝑡 ∧ 𝑦 = ⟨𝑧, 𝑡⟩) ↔ (𝑡 = 𝑧 ∧ 𝑦 = ⟨𝑧, 𝑡⟩))
22		opeq2 4840	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑧 → ⟨𝑧, 𝑡⟩ = ⟨𝑧, 𝑧⟩)
23	22	eqeq2d 2774	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑧 → (𝑦 = ⟨𝑧, 𝑡⟩ ↔ 𝑦 = ⟨𝑧, 𝑧⟩))
24	23	pm5.32i 584	. . . . . . . . . . . 12 ⊢ ((𝑡 = 𝑧 ∧ 𝑦 = ⟨𝑧, 𝑡⟩) ↔ (𝑡 = 𝑧 ∧ 𝑦 = ⟨𝑧, 𝑧⟩))
25	19, 21, 24	3bitri 300	. . . . . . . . . . 11 ⊢ ((𝑦 = ⟨𝑧, 𝑡⟩ ∧ 𝑧 = 𝑡) ↔ (𝑡 = 𝑧 ∧ 𝑦 = ⟨𝑧, 𝑧⟩))
26	25	exbii 1878	. . . . . . . . . 10 ⊢ (∃𝑡(𝑦 = ⟨𝑧, 𝑡⟩ ∧ 𝑧 = 𝑡) ↔ ∃𝑡(𝑡 = 𝑧 ∧ 𝑦 = ⟨𝑧, 𝑧⟩))
27		biidd 265	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑧 → (𝑦 = ⟨𝑧, 𝑧⟩ ↔ 𝑦 = ⟨𝑧, 𝑧⟩))
28	27	equsexvw 2035	. . . . . . . . . 10 ⊢ (∃𝑡(𝑡 = 𝑧 ∧ 𝑦 = ⟨𝑧, 𝑧⟩) ↔ 𝑦 = ⟨𝑧, 𝑧⟩)
29	26, 28	bitri 278	. . . . . . . . 9 ⊢ (∃𝑡(𝑦 = ⟨𝑧, 𝑡⟩ ∧ 𝑧 = 𝑡) ↔ 𝑦 = ⟨𝑧, 𝑧⟩)
30	29	exbii 1878	. . . . . . . 8 ⊢ (∃𝑧∃𝑡(𝑦 = ⟨𝑧, 𝑡⟩ ∧ 𝑧 = 𝑡) ↔ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩)
31	17, 18, 30	3bitrri 301	. . . . . . 7 ⊢ (∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩ ↔ 𝑦 ∈ I )
32	15, 31	anbi12ci 640	. . . . . 6 ⊢ (((1st ‘𝑦) = 𝑥 ∧ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩) ↔ (𝑦 ∈ I ∧ 𝑦1st 𝑥))
33	5, 10, 32	3bitr3ri 305	. . . . 5 ⊢ ((𝑦 ∈ I ∧ 𝑦1st 𝑥) ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩))
34		id 23	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → 𝑧 = 𝑥)
35	34, 34	opeq12d 4847	. . . . . . 7 ⊢ (𝑧 = 𝑥 → ⟨𝑧, 𝑧⟩ = ⟨𝑥, 𝑥⟩)
36	35	eqeq2d 2774	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑦 = ⟨𝑧, 𝑧⟩ ↔ 𝑦 = ⟨𝑥, 𝑥⟩))
37	36	equsexvw 2035	. . . . 5 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩) ↔ 𝑦 = ⟨𝑥, 𝑥⟩)
38	33, 37	bitri 278	. . . 4 ⊢ ((𝑦 ∈ I ∧ 𝑦1st 𝑥) ↔ 𝑦 = ⟨𝑥, 𝑥⟩)
39	3, 4, 38	3bitri 300	. . 3 ⊢ (𝑥◡(1st ↾ I )𝑦 ↔ 𝑦 = ⟨𝑥, 𝑥⟩)
40	39	opabbii 5179	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥◡(1st ↾ I )𝑦} = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = ⟨𝑥, 𝑥⟩}
41		relcnv 6108	. . 3 ⊢ Rel ◡(1st ↾ I )
42		dfrel4v 6190	. . 3 ⊢ (Rel ◡(1st ↾ I ) ↔ ◡(1st ↾ I ) = {⟨𝑥, 𝑦⟩ ∣ 𝑥◡(1st ↾ I )𝑦})
43	41, 42	mpbi 233	. 2 ⊢ ◡(1st ↾ I ) = {⟨𝑥, 𝑦⟩ ∣ 𝑥◡(1st ↾ I )𝑦}
44		mptv 5218	. 2 ⊢ (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = ⟨𝑥, 𝑥⟩}
45	40, 43, 44	3eqtr4i 2796	1 ⊢ ◡(1st ↾ I ) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  Vcvv 3455  ⟨cop 4596   class class class wbr 5110  {copab 5174   ↦ cmpt 5193   I cid 5557  ^◡ccnv 5662   ↾ cres 5665  Rel wrel 5668   Fn wfn 6533  –onto→wfo 6536  ‘cfv 6538  1^st c1st 7985

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 5258  ax-nul 5270  ax-pr 5406  ax-un 7734

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 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fo 6544  df-fv 6546  df-1st 7987

This theorem is referenced by:  fsplitfpar  8114