Theorem fsplitOLD 7958

Description: Obsolete proof of fsplit 7957 as of 31-Dec-2023. (Contributed by NM, 17-Sep-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem fsplitOLD

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

Step	Hyp	Ref	Expression
1		vex 3436	. . . . 5 ⊢ 𝑥 ∈ V
2		vex 3436	. . . . 5 ⊢ 𝑦 ∈ V
3	1, 2	brcnv 5791	. . . 4 ⊢ (𝑥◡(1st ↾ I )𝑦 ↔ 𝑦(1st ↾ I )𝑥)
4	1	brresi 5900	. . . . 5 ⊢ (𝑦(1st ↾ I )𝑥 ↔ (𝑦 ∈ I ∧ 𝑦1st 𝑥))
5		19.42v 1957	. . . . . . 7 ⊢ (∃𝑧((1st ‘𝑦) = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩) ↔ ((1st ‘𝑦) = 𝑥 ∧ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩))
6		vex 3436	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
7	6, 6	op1std 7841	. . . . . . . . . 10 ⊢ (𝑦 = ⟨𝑧, 𝑧⟩ → (1st ‘𝑦) = 𝑧)
8	7	eqeq1d 2740	. . . . . . . . 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 1850	. . . . . . 7 ⊢ (∃𝑧((1st ‘𝑦) = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩) ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩))
11		fo1st 7851	. . . . . . . . . 10 ⊢ 1st :V–onto→V
12		fofn 6690	. . . . . . . . . 10 ⊢ (1st :V–onto→V → 1st Fn V)
13	11, 12	ax-mp 5	. . . . . . . . 9 ⊢ 1st Fn V
14		fnbrfvb 6822	. . . . . . . . 9 ⊢ ((1st Fn V ∧ 𝑦 ∈ V) → ((1st ‘𝑦) = 𝑥 ↔ 𝑦1st 𝑥))
15	13, 2, 14	mp2an 689	. . . . . . . 8 ⊢ ((1st ‘𝑦) = 𝑥 ↔ 𝑦1st 𝑥)
16		dfid2 5491	. . . . . . . . . 10 ⊢ I = {⟨𝑧, 𝑧⟩ ∣ 𝑧 = 𝑧}
17	16	eleq2i 2830	. . . . . . . . 9 ⊢ (𝑦 ∈ I ↔ 𝑦 ∈ {⟨𝑧, 𝑧⟩ ∣ 𝑧 = 𝑧})
18		nfe1 2147	. . . . . . . . . . 11 ⊢ Ⅎ𝑧∃𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧)
19	18	19.9 2198	. . . . . . . . . 10 ⊢ (∃𝑧∃𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧) ↔ ∃𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧))
20		elopab 5440	. . . . . . . . . 10 ⊢ (𝑦 ∈ {⟨𝑧, 𝑧⟩ ∣ 𝑧 = 𝑧} ↔ ∃𝑧∃𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧))
21		equid 2015	. . . . . . . . . . . 12 ⊢ 𝑧 = 𝑧
22	21	biantru 530	. . . . . . . . . . 11 ⊢ (𝑦 = ⟨𝑧, 𝑧⟩ ↔ (𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧))
23	22	exbii 1850	. . . . . . . . . 10 ⊢ (∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩ ↔ ∃𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧))
24	19, 20, 23	3bitr4i 303	. . . . . . . . 9 ⊢ (𝑦 ∈ {⟨𝑧, 𝑧⟩ ∣ 𝑧 = 𝑧} ↔ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩)
25	17, 24	bitr2i 275	. . . . . . . 8 ⊢ (∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩ ↔ 𝑦 ∈ I )
26	15, 25	anbi12ci 628	. . . . . . 7 ⊢ (((1st ‘𝑦) = 𝑥 ∧ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩) ↔ (𝑦 ∈ I ∧ 𝑦1st 𝑥))
27	5, 10, 26	3bitr3ri 302	. . . . . 6 ⊢ ((𝑦 ∈ I ∧ 𝑦1st 𝑥) ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩))
28		id 22	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → 𝑧 = 𝑥)
29	28, 28	opeq12d 4812	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ⟨𝑧, 𝑧⟩ = ⟨𝑥, 𝑥⟩)
30	29	eqeq2d 2749	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑦 = ⟨𝑧, 𝑧⟩ ↔ 𝑦 = ⟨𝑥, 𝑥⟩))
31	30	equsexvw 2008	. . . . . 6 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩) ↔ 𝑦 = ⟨𝑥, 𝑥⟩)
32	27, 31	bitri 274	. . . . 5 ⊢ ((𝑦 ∈ I ∧ 𝑦1st 𝑥) ↔ 𝑦 = ⟨𝑥, 𝑥⟩)
33	4, 32	bitri 274	. . . 4 ⊢ (𝑦(1st ↾ I )𝑥 ↔ 𝑦 = ⟨𝑥, 𝑥⟩)
34	3, 33	bitri 274	. . 3 ⊢ (𝑥◡(1st ↾ I )𝑦 ↔ 𝑦 = ⟨𝑥, 𝑥⟩)
35	34	opabbii 5141	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥◡(1st ↾ I )𝑦} = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = ⟨𝑥, 𝑥⟩}
36		relcnv 6012	. . 3 ⊢ Rel ◡(1st ↾ I )
37		dfrel4v 6093	. . 3 ⊢ (Rel ◡(1st ↾ I ) ↔ ◡(1st ↾ I ) = {⟨𝑥, 𝑦⟩ ∣ 𝑥◡(1st ↾ I )𝑦})
38	36, 37	mpbi 229	. 2 ⊢ ◡(1st ↾ I ) = {⟨𝑥, 𝑦⟩ ∣ 𝑥◡(1st ↾ I )𝑦}
39		mptv 5190	. 2 ⊢ (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = ⟨𝑥, 𝑥⟩}
40	35, 38, 39	3eqtr4i 2776	1 ⊢ ◡(1st ↾ I ) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  Vcvv 3432  ⟨cop 4567   class class class wbr 5074  {copab 5136   ↦ cmpt 5157   I cid 5488  ^◡ccnv 5588   ↾ cres 5591  Rel wrel 5594   Fn wfn 6428  –onto→wfo 6431  ‘cfv 6433  1^st c1st 7829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fo 6439  df-fv 6441  df-1st 7831

This theorem is referenced by: (None)