Theorem fsplit 8002

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 8001 in order to build compound functions such as (𝑥 ∈ (0[,)+∞) ↦ ((√‘𝑥) + (sin‘𝑥))). (Contributed by NM, 17-Sep-2007.) Replace use of dfid2 5509 with df-id 5507. (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 3445	. . . . 5 ⊢ 𝑥 ∈ V
2		vex 3445	. . . . 5 ⊢ 𝑦 ∈ V
3	1, 2	brcnv 5811	. . . 4 ⊢ (𝑥◡(1st ↾ I )𝑦 ↔ 𝑦(1st ↾ I )𝑥)
4	1	brresi 5919	. . . 4 ⊢ (𝑦(1st ↾ I )𝑥 ↔ (𝑦 ∈ I ∧ 𝑦1st 𝑥))
5		19.42v 1956	. . . . . 6 ⊢ (∃𝑧((1st ‘𝑦) = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩) ↔ ((1st ‘𝑦) = 𝑥 ∧ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩))
6		vex 3445	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
7	6, 6	op1std 7886	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑧, 𝑧⟩ → (1st ‘𝑦) = 𝑧)
8	7	eqeq1d 2739	. . . . . . . 8 ⊢ (𝑦 = ⟨𝑧, 𝑧⟩ → ((1st ‘𝑦) = 𝑥 ↔ 𝑧 = 𝑥))
9	8	pm5.32ri 576	. . . . . . 7 ⊢ (((1st ‘𝑦) = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩) ↔ (𝑧 = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩))
10	9	exbii 1849	. . . . . 6 ⊢ (∃𝑧((1st ‘𝑦) = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩) ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩))
11		fo1st 7896	. . . . . . . . 9 ⊢ 1st :V–onto→V
12		fofn 6727	. . . . . . . . 9 ⊢ (1st :V–onto→V → 1st Fn V)
13	11, 12	ax-mp 5	. . . . . . . 8 ⊢ 1st Fn V
14		fnbrfvb 6861	. . . . . . . 8 ⊢ ((1st Fn V ∧ 𝑦 ∈ V) → ((1st ‘𝑦) = 𝑥 ↔ 𝑦1st 𝑥))
15	13, 2, 14	mp2an 689	. . . . . . 7 ⊢ ((1st ‘𝑦) = 𝑥 ↔ 𝑦1st 𝑥)
16		df-id 5507	. . . . . . . . 9 ⊢ I = {⟨𝑧, 𝑡⟩ ∣ 𝑧 = 𝑡}
17	16	eleq2i 2829	. . . . . . . 8 ⊢ (𝑦 ∈ I ↔ 𝑦 ∈ {⟨𝑧, 𝑡⟩ ∣ 𝑧 = 𝑡})
18		elopab 5460	. . . . . . . 8 ⊢ (𝑦 ∈ {⟨𝑧, 𝑡⟩ ∣ 𝑧 = 𝑡} ↔ ∃𝑧∃𝑡(𝑦 = ⟨𝑧, 𝑡⟩ ∧ 𝑧 = 𝑡))
19		ancom 461	. . . . . . . . . . . 12 ⊢ ((𝑦 = ⟨𝑧, 𝑡⟩ ∧ 𝑧 = 𝑡) ↔ (𝑧 = 𝑡 ∧ 𝑦 = ⟨𝑧, 𝑡⟩))
20		equcom 2020	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑡 ↔ 𝑡 = 𝑧)
21	20	anbi1i 624	. . . . . . . . . . . 12 ⊢ ((𝑧 = 𝑡 ∧ 𝑦 = ⟨𝑧, 𝑡⟩) ↔ (𝑡 = 𝑧 ∧ 𝑦 = ⟨𝑧, 𝑡⟩))
22		opeq2 4816	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑧 → ⟨𝑧, 𝑡⟩ = ⟨𝑧, 𝑧⟩)
23	22	eqeq2d 2748	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑧 → (𝑦 = ⟨𝑧, 𝑡⟩ ↔ 𝑦 = ⟨𝑧, 𝑧⟩))
24	23	pm5.32i 575	. . . . . . . . . . . 12 ⊢ ((𝑡 = 𝑧 ∧ 𝑦 = ⟨𝑧, 𝑡⟩) ↔ (𝑡 = 𝑧 ∧ 𝑦 = ⟨𝑧, 𝑧⟩))
25	19, 21, 24	3bitri 296	. . . . . . . . . . 11 ⊢ ((𝑦 = ⟨𝑧, 𝑡⟩ ∧ 𝑧 = 𝑡) ↔ (𝑡 = 𝑧 ∧ 𝑦 = ⟨𝑧, 𝑧⟩))
26	25	exbii 1849	. . . . . . . . . 10 ⊢ (∃𝑡(𝑦 = ⟨𝑧, 𝑡⟩ ∧ 𝑧 = 𝑡) ↔ ∃𝑡(𝑡 = 𝑧 ∧ 𝑦 = ⟨𝑧, 𝑧⟩))
27		biidd 261	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑧 → (𝑦 = ⟨𝑧, 𝑧⟩ ↔ 𝑦 = ⟨𝑧, 𝑧⟩))
28	27	equsexvw 2007	. . . . . . . . . 10 ⊢ (∃𝑡(𝑡 = 𝑧 ∧ 𝑦 = ⟨𝑧, 𝑧⟩) ↔ 𝑦 = ⟨𝑧, 𝑧⟩)
29	26, 28	bitri 274	. . . . . . . . 9 ⊢ (∃𝑡(𝑦 = ⟨𝑧, 𝑡⟩ ∧ 𝑧 = 𝑡) ↔ 𝑦 = ⟨𝑧, 𝑧⟩)
30	29	exbii 1849	. . . . . . . 8 ⊢ (∃𝑧∃𝑡(𝑦 = ⟨𝑧, 𝑡⟩ ∧ 𝑧 = 𝑡) ↔ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩)
31	17, 18, 30	3bitrri 297	. . . . . . 7 ⊢ (∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩ ↔ 𝑦 ∈ I )
32	15, 31	anbi12ci 628	. . . . . 6 ⊢ (((1st ‘𝑦) = 𝑥 ∧ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩) ↔ (𝑦 ∈ I ∧ 𝑦1st 𝑥))
33	5, 10, 32	3bitr3ri 301	. . . . 5 ⊢ ((𝑦 ∈ I ∧ 𝑦1st 𝑥) ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩))
34		id 22	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → 𝑧 = 𝑥)
35	34, 34	opeq12d 4823	. . . . . . 7 ⊢ (𝑧 = 𝑥 → ⟨𝑧, 𝑧⟩ = ⟨𝑥, 𝑥⟩)
36	35	eqeq2d 2748	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑦 = ⟨𝑧, 𝑧⟩ ↔ 𝑦 = ⟨𝑥, 𝑥⟩))
37	36	equsexvw 2007	. . . . 5 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩) ↔ 𝑦 = ⟨𝑥, 𝑥⟩)
38	33, 37	bitri 274	. . . 4 ⊢ ((𝑦 ∈ I ∧ 𝑦1st 𝑥) ↔ 𝑦 = ⟨𝑥, 𝑥⟩)
39	3, 4, 38	3bitri 296	. . 3 ⊢ (𝑥◡(1st ↾ I )𝑦 ↔ 𝑦 = ⟨𝑥, 𝑥⟩)
40	39	opabbii 5154	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥◡(1st ↾ I )𝑦} = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = ⟨𝑥, 𝑥⟩}
41		relcnv 6029	. . 3 ⊢ Rel ◡(1st ↾ I )
42		dfrel4v 6115	. . 3 ⊢ (Rel ◡(1st ↾ I ) ↔ ◡(1st ↾ I ) = {⟨𝑥, 𝑦⟩ ∣ 𝑥◡(1st ↾ I )𝑦})
43	41, 42	mpbi 229	. 2 ⊢ ◡(1st ↾ I ) = {⟨𝑥, 𝑦⟩ ∣ 𝑥◡(1st ↾ I )𝑦}
44		mptv 5203	. 2 ⊢ (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = ⟨𝑥, 𝑥⟩}
45	40, 43, 44	3eqtr4i 2775	1 ⊢ ◡(1st ↾ I ) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1540  ∃wex 1780   ∈ wcel 2105  Vcvv 3441  ⟨cop 4577   class class class wbr 5087  {copab 5149   ↦ cmpt 5170   I cid 5506  ^◡ccnv 5606   ↾ cres 5609  Rel wrel 5612   Fn wfn 6460  –onto→wfo 6463  ‘cfv 6465  1^st c1st 7874

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-sep 5238  ax-nul 5245  ax-pr 5367  ax-un 7628

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3443  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4268  df-if 4472  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4851  df-br 5088  df-opab 5150  df-mpt 5171  df-id 5507  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-fo 6471  df-fv 6473  df-1st 7876

This theorem is referenced by:  fsplitfpar  8003