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Theorem fsplitOLD 7812
Description: Obsolete proof of fsplit 7811 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.
StepHypRef Expression
1 vex 3497 . . . . 5 𝑥 ∈ V
2 vex 3497 . . . . 5 𝑦 ∈ V
31, 2brcnv 5752 . . . 4 (𝑥(1st ↾ I )𝑦𝑦(1st ↾ I )𝑥)
41brresi 5861 . . . . 5 (𝑦(1st ↾ I )𝑥 ↔ (𝑦 ∈ I ∧ 𝑦1st 𝑥))
5 19.42v 1950 . . . . . . 7 (∃𝑧((1st𝑦) = 𝑥𝑦 = ⟨𝑧, 𝑧⟩) ↔ ((1st𝑦) = 𝑥 ∧ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩))
6 vex 3497 . . . . . . . . . . 11 𝑧 ∈ V
76, 6op1std 7698 . . . . . . . . . 10 (𝑦 = ⟨𝑧, 𝑧⟩ → (1st𝑦) = 𝑧)
87eqeq1d 2823 . . . . . . . . 9 (𝑦 = ⟨𝑧, 𝑧⟩ → ((1st𝑦) = 𝑥𝑧 = 𝑥))
98pm5.32ri 578 . . . . . . . 8 (((1st𝑦) = 𝑥𝑦 = ⟨𝑧, 𝑧⟩) ↔ (𝑧 = 𝑥𝑦 = ⟨𝑧, 𝑧⟩))
109exbii 1844 . . . . . . 7 (∃𝑧((1st𝑦) = 𝑥𝑦 = ⟨𝑧, 𝑧⟩) ↔ ∃𝑧(𝑧 = 𝑥𝑦 = ⟨𝑧, 𝑧⟩))
11 fo1st 7708 . . . . . . . . . 10 1st :V–onto→V
12 fofn 6591 . . . . . . . . . 10 (1st :V–onto→V → 1st Fn V)
1311, 12ax-mp 5 . . . . . . . . 9 1st Fn V
14 fnbrfvb 6717 . . . . . . . . 9 ((1st Fn V ∧ 𝑦 ∈ V) → ((1st𝑦) = 𝑥𝑦1st 𝑥))
1513, 2, 14mp2an 690 . . . . . . . 8 ((1st𝑦) = 𝑥𝑦1st 𝑥)
16 dfid2 5462 . . . . . . . . . 10 I = {⟨𝑧, 𝑧⟩ ∣ 𝑧 = 𝑧}
1716eleq2i 2904 . . . . . . . . 9 (𝑦 ∈ I ↔ 𝑦 ∈ {⟨𝑧, 𝑧⟩ ∣ 𝑧 = 𝑧})
18 nfe1 2150 . . . . . . . . . . 11 𝑧𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧)
191819.9 2201 . . . . . . . . . 10 (∃𝑧𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧) ↔ ∃𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧))
20 elopab 5413 . . . . . . . . . 10 (𝑦 ∈ {⟨𝑧, 𝑧⟩ ∣ 𝑧 = 𝑧} ↔ ∃𝑧𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧))
21 equid 2015 . . . . . . . . . . . 12 𝑧 = 𝑧
2221biantru 532 . . . . . . . . . . 11 (𝑦 = ⟨𝑧, 𝑧⟩ ↔ (𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧))
2322exbii 1844 . . . . . . . . . 10 (∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩ ↔ ∃𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧))
2419, 20, 233bitr4i 305 . . . . . . . . 9 (𝑦 ∈ {⟨𝑧, 𝑧⟩ ∣ 𝑧 = 𝑧} ↔ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩)
2517, 24bitr2i 278 . . . . . . . 8 (∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩ ↔ 𝑦 ∈ I )
2615, 25anbi12ci 629 . . . . . . 7 (((1st𝑦) = 𝑥 ∧ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩) ↔ (𝑦 ∈ I ∧ 𝑦1st 𝑥))
275, 10, 263bitr3ri 304 . . . . . 6 ((𝑦 ∈ I ∧ 𝑦1st 𝑥) ↔ ∃𝑧(𝑧 = 𝑥𝑦 = ⟨𝑧, 𝑧⟩))
28 id 22 . . . . . . . . 9 (𝑧 = 𝑥𝑧 = 𝑥)
2928, 28opeq12d 4810 . . . . . . . 8 (𝑧 = 𝑥 → ⟨𝑧, 𝑧⟩ = ⟨𝑥, 𝑥⟩)
3029eqeq2d 2832 . . . . . . 7 (𝑧 = 𝑥 → (𝑦 = ⟨𝑧, 𝑧⟩ ↔ 𝑦 = ⟨𝑥, 𝑥⟩))
3130equsexvw 2007 . . . . . 6 (∃𝑧(𝑧 = 𝑥𝑦 = ⟨𝑧, 𝑧⟩) ↔ 𝑦 = ⟨𝑥, 𝑥⟩)
3227, 31bitri 277 . . . . 5 ((𝑦 ∈ I ∧ 𝑦1st 𝑥) ↔ 𝑦 = ⟨𝑥, 𝑥⟩)
334, 32bitri 277 . . . 4 (𝑦(1st ↾ I )𝑥𝑦 = ⟨𝑥, 𝑥⟩)
343, 33bitri 277 . . 3 (𝑥(1st ↾ I )𝑦𝑦 = ⟨𝑥, 𝑥⟩)
3534opabbii 5132 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑥(1st ↾ I )𝑦} = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = ⟨𝑥, 𝑥⟩}
36 relcnv 5966 . . 3 Rel (1st ↾ I )
37 dfrel4v 6046 . . 3 (Rel (1st ↾ I ) ↔ (1st ↾ I ) = {⟨𝑥, 𝑦⟩ ∣ 𝑥(1st ↾ I )𝑦})
3836, 37mpbi 232 . 2 (1st ↾ I ) = {⟨𝑥, 𝑦⟩ ∣ 𝑥(1st ↾ I )𝑦}
39 mptv 5170 . 2 (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = ⟨𝑥, 𝑥⟩}
4035, 38, 393eqtr4i 2854 1 (1st ↾ I ) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398   = wceq 1533  wex 1776  wcel 2110  Vcvv 3494  cop 4572   class class class wbr 5065  {copab 5127  cmpt 5145   I cid 5458  ccnv 5553  cres 5556  Rel wrel 5559   Fn wfn 6349  ontowfo 6352  cfv 6354  1st c1st 7686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-13 2386  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fo 6360  df-fv 6362  df-1st 7688
This theorem is referenced by: (None)
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