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Theorem fsplit 7267
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 7266 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 3198 . . . . 5 𝑥 ∈ V
2 vex 3198 . . . . 5 𝑦 ∈ V
31, 2brcnv 5294 . . . 4 (𝑥(1st ↾ I )𝑦𝑦(1st ↾ I )𝑥)
41brres 5391 . . . . 5 (𝑦(1st ↾ I )𝑥 ↔ (𝑦1st 𝑥𝑦 ∈ I ))
5 19.42v 1916 . . . . . . 7 (∃𝑧((1st𝑦) = 𝑥𝑦 = ⟨𝑧, 𝑧⟩) ↔ ((1st𝑦) = 𝑥 ∧ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩))
6 vex 3198 . . . . . . . . . . 11 𝑧 ∈ V
76, 6op1std 7163 . . . . . . . . . 10 (𝑦 = ⟨𝑧, 𝑧⟩ → (1st𝑦) = 𝑧)
87eqeq1d 2622 . . . . . . . . 9 (𝑦 = ⟨𝑧, 𝑧⟩ → ((1st𝑦) = 𝑥𝑧 = 𝑥))
98pm5.32ri 669 . . . . . . . 8 (((1st𝑦) = 𝑥𝑦 = ⟨𝑧, 𝑧⟩) ↔ (𝑧 = 𝑥𝑦 = ⟨𝑧, 𝑧⟩))
109exbii 1772 . . . . . . 7 (∃𝑧((1st𝑦) = 𝑥𝑦 = ⟨𝑧, 𝑧⟩) ↔ ∃𝑧(𝑧 = 𝑥𝑦 = ⟨𝑧, 𝑧⟩))
11 fo1st 7173 . . . . . . . . . 10 1st :V–onto→V
12 fofn 6104 . . . . . . . . . 10 (1st :V–onto→V → 1st Fn V)
1311, 12ax-mp 5 . . . . . . . . 9 1st Fn V
14 fnbrfvb 6223 . . . . . . . . 9 ((1st Fn V ∧ 𝑦 ∈ V) → ((1st𝑦) = 𝑥𝑦1st 𝑥))
1513, 2, 14mp2an 707 . . . . . . . 8 ((1st𝑦) = 𝑥𝑦1st 𝑥)
16 dfid2 5017 . . . . . . . . . 10 I = {⟨𝑧, 𝑧⟩ ∣ 𝑧 = 𝑧}
1716eleq2i 2691 . . . . . . . . 9 (𝑦 ∈ I ↔ 𝑦 ∈ {⟨𝑧, 𝑧⟩ ∣ 𝑧 = 𝑧})
18 nfe1 2025 . . . . . . . . . . 11 𝑧𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧)
191819.9 2070 . . . . . . . . . 10 (∃𝑧𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧) ↔ ∃𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧))
20 elopab 4973 . . . . . . . . . 10 (𝑦 ∈ {⟨𝑧, 𝑧⟩ ∣ 𝑧 = 𝑧} ↔ ∃𝑧𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧))
21 equid 1937 . . . . . . . . . . . 12 𝑧 = 𝑧
2221biantru 526 . . . . . . . . . . 11 (𝑦 = ⟨𝑧, 𝑧⟩ ↔ (𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧))
2322exbii 1772 . . . . . . . . . 10 (∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩ ↔ ∃𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧))
2419, 20, 233bitr4i 292 . . . . . . . . 9 (𝑦 ∈ {⟨𝑧, 𝑧⟩ ∣ 𝑧 = 𝑧} ↔ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩)
2517, 24bitr2i 265 . . . . . . . 8 (∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩ ↔ 𝑦 ∈ I )
2615, 25anbi12i 732 . . . . . . 7 (((1st𝑦) = 𝑥 ∧ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩) ↔ (𝑦1st 𝑥𝑦 ∈ I ))
275, 10, 263bitr3ri 291 . . . . . 6 ((𝑦1st 𝑥𝑦 ∈ I ) ↔ ∃𝑧(𝑧 = 𝑥𝑦 = ⟨𝑧, 𝑧⟩))
28 id 22 . . . . . . . . 9 (𝑧 = 𝑥𝑧 = 𝑥)
2928, 28opeq12d 4401 . . . . . . . 8 (𝑧 = 𝑥 → ⟨𝑧, 𝑧⟩ = ⟨𝑥, 𝑥⟩)
3029eqeq2d 2630 . . . . . . 7 (𝑧 = 𝑥 → (𝑦 = ⟨𝑧, 𝑧⟩ ↔ 𝑦 = ⟨𝑥, 𝑥⟩))
311, 30ceqsexv 3237 . . . . . 6 (∃𝑧(𝑧 = 𝑥𝑦 = ⟨𝑧, 𝑧⟩) ↔ 𝑦 = ⟨𝑥, 𝑥⟩)
3227, 31bitri 264 . . . . 5 ((𝑦1st 𝑥𝑦 ∈ I ) ↔ 𝑦 = ⟨𝑥, 𝑥⟩)
334, 32bitri 264 . . . 4 (𝑦(1st ↾ I )𝑥𝑦 = ⟨𝑥, 𝑥⟩)
343, 33bitri 264 . . 3 (𝑥(1st ↾ I )𝑦𝑦 = ⟨𝑥, 𝑥⟩)
3534opabbii 4708 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑥(1st ↾ I )𝑦} = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = ⟨𝑥, 𝑥⟩}
36 relcnv 5491 . . 3 Rel (1st ↾ I )
37 dfrel4v 5572 . . 3 (Rel (1st ↾ I ) ↔ (1st ↾ I ) = {⟨𝑥, 𝑦⟩ ∣ 𝑥(1st ↾ I )𝑦})
3836, 37mpbi 220 . 2 (1st ↾ I ) = {⟨𝑥, 𝑦⟩ ∣ 𝑥(1st ↾ I )𝑦}
39 mptv 4742 . 2 (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = ⟨𝑥, 𝑥⟩}
4035, 38, 393eqtr4i 2652 1 (1st ↾ I ) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1481  wex 1702  wcel 1988  Vcvv 3195  cop 4174   class class class wbr 4644  {copab 4703  cmpt 4720   I cid 5013  ccnv 5103  cres 5106  Rel wrel 5109   Fn wfn 5871  ontowfo 5874  cfv 5876  1st c1st 7151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fo 5882  df-fv 5884  df-1st 7153
This theorem is referenced by: (None)
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