Theorem fsplitOLD 7929

Description: Obsolete proof of fsplit 7928 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 3426	. . . . 5 ⊢ 𝑥 ∈ V
2		vex 3426	. . . . 5 ⊢ 𝑦 ∈ V
3	1, 2	brcnv 5780	. . . 4 ⊢ (𝑥◡(1st ↾ I )𝑦 ↔ 𝑦(1st ↾ I )𝑥)
4	1	brresi 5889	. . . . 5 ⊢ (𝑦(1st ↾ I )𝑥 ↔ (𝑦 ∈ I ∧ 𝑦1st 𝑥))
5		19.42v 1958	. . . . . . 7 ⊢ (∃𝑧((1st ‘𝑦) = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩) ↔ ((1st ‘𝑦) = 𝑥 ∧ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩))
6		vex 3426	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
7	6, 6	op1std 7814	. . . . . . . . . 10 ⊢ (𝑦 = ⟨𝑧, 𝑧⟩ → (1st ‘𝑦) = 𝑧)
8	7	eqeq1d 2740	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑧, 𝑧⟩ → ((1st ‘𝑦) = 𝑥 ↔ 𝑧 = 𝑥))
9	8	pm5.32ri 575	. . . . . . . 8 ⊢ (((1st ‘𝑦) = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩) ↔ (𝑧 = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩))
10	9	exbii 1851	. . . . . . 7 ⊢ (∃𝑧((1st ‘𝑦) = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩) ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩))
11		fo1st 7824	. . . . . . . . . 10 ⊢ 1st :V–onto→V
12		fofn 6674	. . . . . . . . . 10 ⊢ (1st :V–onto→V → 1st Fn V)
13	11, 12	ax-mp 5	. . . . . . . . 9 ⊢ 1st Fn V
14		fnbrfvb 6804	. . . . . . . . 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 5482	. . . . . . . . . 10 ⊢ I = {⟨𝑧, 𝑧⟩ ∣ 𝑧 = 𝑧}
17	16	eleq2i 2830	. . . . . . . . 9 ⊢ (𝑦 ∈ I ↔ 𝑦 ∈ {⟨𝑧, 𝑧⟩ ∣ 𝑧 = 𝑧})
18		nfe1 2149	. . . . . . . . . . 11 ⊢ Ⅎ𝑧∃𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧)
19	18	19.9 2201	. . . . . . . . . 10 ⊢ (∃𝑧∃𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧) ↔ ∃𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧))
20		elopab 5433	. . . . . . . . . 10 ⊢ (𝑦 ∈ {⟨𝑧, 𝑧⟩ ∣ 𝑧 = 𝑧} ↔ ∃𝑧∃𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧))
21		equid 2016	. . . . . . . . . . . 12 ⊢ 𝑧 = 𝑧
22	21	biantru 529	. . . . . . . . . . 11 ⊢ (𝑦 = ⟨𝑧, 𝑧⟩ ↔ (𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧))
23	22	exbii 1851	. . . . . . . . . 10 ⊢ (∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩ ↔ ∃𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧))
24	19, 20, 23	3bitr4i 302	. . . . . . . . 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 627	. . . . . . 7 ⊢ (((1st ‘𝑦) = 𝑥 ∧ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩) ↔ (𝑦 ∈ I ∧ 𝑦1st 𝑥))
27	5, 10, 26	3bitr3ri 301	. . . . . 6 ⊢ ((𝑦 ∈ I ∧ 𝑦1st 𝑥) ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑦 = ⟨𝑧, 𝑧⟩))
28		id 22	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → 𝑧 = 𝑥)
29	28, 28	opeq12d 4809	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ⟨𝑧, 𝑧⟩ = ⟨𝑥, 𝑥⟩)
30	29	eqeq2d 2749	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑦 = ⟨𝑧, 𝑧⟩ ↔ 𝑦 = ⟨𝑥, 𝑥⟩))
31	30	equsexvw 2009	. . . . . 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 5137	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥◡(1st ↾ I )𝑦} = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = ⟨𝑥, 𝑥⟩}
36		relcnv 6001	. . 3 ⊢ Rel ◡(1st ↾ I )
37		dfrel4v 6082	. . 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 5186	. 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 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  Vcvv 3422  ⟨cop 4564   class class class wbr 5070  {copab 5132   ↦ cmpt 5153   I cid 5479  ^◡ccnv 5579   ↾ cres 5582  Rel wrel 5585   Fn wfn 6413  –onto→wfo 6416  ‘cfv 6418  1^st c1st 7802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fo 6424  df-fv 6426  df-1st 7804

This theorem is referenced by: (None)