Theorem fsplit 7675

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 7674 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 3443	. . . . 5 ⊢ 𝑥 ∈ V
2		vex 3443	. . . . 5 ⊢ 𝑦 ∈ V
3	1, 2	brcnv 5646	. . . 4 ⊢ (𝑥◡(1st ↾ I )𝑦 ↔ 𝑦(1st ↾ I )𝑥)
4	1	brresi 5750	. . . . 5 ⊢ (𝑦(1st ↾ I )𝑥 ↔ (𝑦 ∈ I ∧ 𝑦1st 𝑥))
5		19.42v 1935	. . . . . . 7 ⊢ (∃𝑧((1st ‘𝑦) = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩) ↔ ((1st ‘𝑦) = 𝑥 ∧ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩))
6		vex 3443	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
7	6, 6	op1std 7562	. . . . . . . . . 10 ⊢ (𝑦 = ⟨𝑧, 𝑧⟩ → (1st ‘𝑦) = 𝑧)
8	7	eqeq1d 2799	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑧, 𝑧⟩ → ((1st ‘𝑦) = 𝑥 ↔ 𝑧 = 𝑥))
9	8	pm5.32ri 576	. . . . . . . 8 ⊢ (((1st ‘𝑦) = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩) ↔ (𝑧 = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩))
10	9	exbii 1833	. . . . . . 7 ⊢ (∃𝑧((1st ‘𝑦) = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩) ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩))
11		fo1st 7572	. . . . . . . . . 10 ⊢ 1st :V–onto→V
12		fofn 6467	. . . . . . . . . 10 ⊢ (1st :V–onto→V → 1st Fn V)
13	11, 12	ax-mp 5	. . . . . . . . 9 ⊢ 1st Fn V
14		fnbrfvb 6593	. . . . . . . . 9 ⊢ ((1st Fn V ∧ 𝑦 ∈ V) → ((1st ‘𝑦) = 𝑥 ↔ 𝑦1st 𝑥))
15	13, 2, 14	mp2an 688	. . . . . . . 8 ⊢ ((1st ‘𝑦) = 𝑥 ↔ 𝑦1st 𝑥)
16		dfid2 5358	. . . . . . . . . 10 ⊢ I = {⟨𝑧, 𝑧⟩ ∣ 𝑧 = 𝑧}
17	16	eleq2i 2876	. . . . . . . . 9 ⊢ (𝑦 ∈ I ↔ 𝑦 ∈ {⟨𝑧, 𝑧⟩ ∣ 𝑧 = 𝑧})
18		nfe1 2122	. . . . . . . . . . 11 ⊢ Ⅎ𝑧∃𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧)
19	18	19.9 2172	. . . . . . . . . 10 ⊢ (∃𝑧∃𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧) ↔ ∃𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧))
20		elopab 5311	. . . . . . . . . 10 ⊢ (𝑦 ∈ {⟨𝑧, 𝑧⟩ ∣ 𝑧 = 𝑧} ↔ ∃𝑧∃𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧))
21		equid 2000	. . . . . . . . . . . 12 ⊢ 𝑧 = 𝑧
22	21	biantru 530	. . . . . . . . . . 11 ⊢ (𝑦 = ⟨𝑧, 𝑧⟩ ↔ (𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧))
23	22	exbii 1833	. . . . . . . . . 10 ⊢ (∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩ ↔ ∃𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧))
24	19, 20, 23	3bitr4i 304	. . . . . . . . 9 ⊢ (𝑦 ∈ {⟨𝑧, 𝑧⟩ ∣ 𝑧 = 𝑧} ↔ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩)
25	17, 24	bitr2i 277	. . . . . . . 8 ⊢ (∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩ ↔ 𝑦 ∈ I )
26	15, 25	anbi12ci 627	. . . . . . 7 ⊢ (((1st ‘𝑦) = 𝑥 ∧ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩) ↔ (𝑦 ∈ I ∧ 𝑦1st 𝑥))
27	5, 10, 26	3bitr3ri 303	. . . . . 6 ⊢ ((𝑦 ∈ I ∧ 𝑦1st 𝑥) ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩))
28		id 22	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → 𝑧 = 𝑥)
29	28, 28	opeq12d 4724	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ⟨𝑧, 𝑧⟩ = ⟨𝑥, 𝑥⟩)
30	29	eqeq2d 2807	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑦 = ⟨𝑧, 𝑧⟩ ↔ 𝑦 = ⟨𝑥, 𝑥⟩))
31	30	equsexvw 1992	. . . . . 6 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩) ↔ 𝑦 = ⟨𝑥, 𝑥⟩)
32	27, 31	bitri 276	. . . . 5 ⊢ ((𝑦 ∈ I ∧ 𝑦1st 𝑥) ↔ 𝑦 = ⟨𝑥, 𝑥⟩)
33	4, 32	bitri 276	. . . 4 ⊢ (𝑦(1st ↾ I )𝑥 ↔ 𝑦 = ⟨𝑥, 𝑥⟩)
34	3, 33	bitri 276	. . 3 ⊢ (𝑥◡(1st ↾ I )𝑦 ↔ 𝑦 = ⟨𝑥, 𝑥⟩)
35	34	opabbii 5035	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥◡(1st ↾ I )𝑦} = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = ⟨𝑥, 𝑥⟩}
36		relcnv 5850	. . 3 ⊢ Rel ◡(1st ↾ I )
37		dfrel4v 5930	. . 3 ⊢ (Rel ◡(1st ↾ I ) ↔ ◡(1st ↾ I ) = {⟨𝑥, 𝑦⟩ ∣ 𝑥◡(1st ↾ I )𝑦})
38	36, 37	mpbi 231	. 2 ⊢ ◡(1st ↾ I ) = {⟨𝑥, 𝑦⟩ ∣ 𝑥◡(1st ↾ I )𝑦}
39		mptv 5069	. 2 ⊢ (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = ⟨𝑥, 𝑥⟩}
40	35, 38, 39	3eqtr4i 2831	1 ⊢ ◡(1st ↾ I ) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)

Syntax hints:   ↔ wb 207   ∧ wa 396   = wceq 1525  ∃wex 1765   ∈ wcel 2083  Vcvv 3440  ⟨cop 4484   class class class wbr 4968  {copab 5030   ↦ cmpt 5047   I cid 5354  ^◡ccnv 5449   ↾ cres 5452  Rel wrel 5455   Fn wfn 6227  –onto→wfo 6230  ‘cfv 6232  1^st c1st 7550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-fo 6238  df-fv 6240  df-1st 7552

This theorem is referenced by: (None)