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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.
StepHypRef Expression
1 vex 3443 . . . . 5 𝑥 ∈ V
2 vex 3443 . . . . 5 𝑦 ∈ V
31, 2brcnv 5646 . . . 4 (𝑥(1st ↾ I )𝑦𝑦(1st ↾ I )𝑥)
41brresi 5750 . . . . 5 (𝑦(1st ↾ I )𝑥 ↔ (𝑦 ∈ I ∧ 𝑦1st 𝑥))
5 19.42v 1935 . . . . . . 7 (∃𝑧((1st𝑦) = 𝑥𝑦 = ⟨𝑧, 𝑧⟩) ↔ ((1st𝑦) = 𝑥 ∧ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩))
6 vex 3443 . . . . . . . . . . 11 𝑧 ∈ V
76, 6op1std 7562 . . . . . . . . . 10 (𝑦 = ⟨𝑧, 𝑧⟩ → (1st𝑦) = 𝑧)
87eqeq1d 2799 . . . . . . . . 9 (𝑦 = ⟨𝑧, 𝑧⟩ → ((1st𝑦) = 𝑥𝑧 = 𝑥))
98pm5.32ri 576 . . . . . . . 8 (((1st𝑦) = 𝑥𝑦 = ⟨𝑧, 𝑧⟩) ↔ (𝑧 = 𝑥𝑦 = ⟨𝑧, 𝑧⟩))
109exbii 1833 . . . . . . 7 (∃𝑧((1st𝑦) = 𝑥𝑦 = ⟨𝑧, 𝑧⟩) ↔ ∃𝑧(𝑧 = 𝑥𝑦 = ⟨𝑧, 𝑧⟩))
11 fo1st 7572 . . . . . . . . . 10 1st :V–onto→V
12 fofn 6467 . . . . . . . . . 10 (1st :V–onto→V → 1st Fn V)
1311, 12ax-mp 5 . . . . . . . . 9 1st Fn V
14 fnbrfvb 6593 . . . . . . . . 9 ((1st Fn V ∧ 𝑦 ∈ V) → ((1st𝑦) = 𝑥𝑦1st 𝑥))
1513, 2, 14mp2an 688 . . . . . . . 8 ((1st𝑦) = 𝑥𝑦1st 𝑥)
16 dfid2 5358 . . . . . . . . . 10 I = {⟨𝑧, 𝑧⟩ ∣ 𝑧 = 𝑧}
1716eleq2i 2876 . . . . . . . . 9 (𝑦 ∈ I ↔ 𝑦 ∈ {⟨𝑧, 𝑧⟩ ∣ 𝑧 = 𝑧})
18 nfe1 2122 . . . . . . . . . . 11 𝑧𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧)
191819.9 2172 . . . . . . . . . 10 (∃𝑧𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧) ↔ ∃𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧))
20 elopab 5311 . . . . . . . . . 10 (𝑦 ∈ {⟨𝑧, 𝑧⟩ ∣ 𝑧 = 𝑧} ↔ ∃𝑧𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧))
21 equid 2000 . . . . . . . . . . . 12 𝑧 = 𝑧
2221biantru 530 . . . . . . . . . . 11 (𝑦 = ⟨𝑧, 𝑧⟩ ↔ (𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧))
2322exbii 1833 . . . . . . . . . 10 (∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩ ↔ ∃𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧))
2419, 20, 233bitr4i 304 . . . . . . . . 9 (𝑦 ∈ {⟨𝑧, 𝑧⟩ ∣ 𝑧 = 𝑧} ↔ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩)
2517, 24bitr2i 277 . . . . . . . 8 (∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩ ↔ 𝑦 ∈ I )
2615, 25anbi12ci 627 . . . . . . 7 (((1st𝑦) = 𝑥 ∧ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩) ↔ (𝑦 ∈ I ∧ 𝑦1st 𝑥))
275, 10, 263bitr3ri 303 . . . . . 6 ((𝑦 ∈ I ∧ 𝑦1st 𝑥) ↔ ∃𝑧(𝑧 = 𝑥𝑦 = ⟨𝑧, 𝑧⟩))
28 id 22 . . . . . . . . 9 (𝑧 = 𝑥𝑧 = 𝑥)
2928, 28opeq12d 4724 . . . . . . . 8 (𝑧 = 𝑥 → ⟨𝑧, 𝑧⟩ = ⟨𝑥, 𝑥⟩)
3029eqeq2d 2807 . . . . . . 7 (𝑧 = 𝑥 → (𝑦 = ⟨𝑧, 𝑧⟩ ↔ 𝑦 = ⟨𝑥, 𝑥⟩))
3130equsexvw 1992 . . . . . 6 (∃𝑧(𝑧 = 𝑥𝑦 = ⟨𝑧, 𝑧⟩) ↔ 𝑦 = ⟨𝑥, 𝑥⟩)
3227, 31bitri 276 . . . . 5 ((𝑦 ∈ I ∧ 𝑦1st 𝑥) ↔ 𝑦 = ⟨𝑥, 𝑥⟩)
334, 32bitri 276 . . . 4 (𝑦(1st ↾ I )𝑥𝑦 = ⟨𝑥, 𝑥⟩)
343, 33bitri 276 . . 3 (𝑥(1st ↾ I )𝑦𝑦 = ⟨𝑥, 𝑥⟩)
3534opabbii 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})
3836, 37mpbi 231 . 2 (1st ↾ I ) = {⟨𝑥, 𝑦⟩ ∣ 𝑥(1st ↾ I )𝑦}
39 mptv 5069 . 2 (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = ⟨𝑥, 𝑥⟩}
4035, 38, 393eqtr4i 2831 1 (1st ↾ I ) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)
Colors of variables: wff setvar class
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  ontowfo 6230  cfv 6232  1st 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)
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