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Theorem fsplit 7230
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 7229 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 3189 . . . . 5 𝑥 ∈ V
2 vex 3189 . . . . 5 𝑦 ∈ V
31, 2brcnv 5267 . . . 4 (𝑥(1st ↾ I )𝑦𝑦(1st ↾ I )𝑥)
41brres 5364 . . . . 5 (𝑦(1st ↾ I )𝑥 ↔ (𝑦1st 𝑥𝑦 ∈ I ))
5 19.42v 1915 . . . . . . 7 (∃𝑧((1st𝑦) = 𝑥𝑦 = ⟨𝑧, 𝑧⟩) ↔ ((1st𝑦) = 𝑥 ∧ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩))
6 vex 3189 . . . . . . . . . . 11 𝑧 ∈ V
76, 6op1std 7126 . . . . . . . . . 10 (𝑦 = ⟨𝑧, 𝑧⟩ → (1st𝑦) = 𝑧)
87eqeq1d 2623 . . . . . . . . 9 (𝑦 = ⟨𝑧, 𝑧⟩ → ((1st𝑦) = 𝑥𝑧 = 𝑥))
98pm5.32ri 669 . . . . . . . 8 (((1st𝑦) = 𝑥𝑦 = ⟨𝑧, 𝑧⟩) ↔ (𝑧 = 𝑥𝑦 = ⟨𝑧, 𝑧⟩))
109exbii 1771 . . . . . . 7 (∃𝑧((1st𝑦) = 𝑥𝑦 = ⟨𝑧, 𝑧⟩) ↔ ∃𝑧(𝑧 = 𝑥𝑦 = ⟨𝑧, 𝑧⟩))
11 fo1st 7136 . . . . . . . . . 10 1st :V–onto→V
12 fofn 6076 . . . . . . . . . 10 (1st :V–onto→V → 1st Fn V)
1311, 12ax-mp 5 . . . . . . . . 9 1st Fn V
14 fnbrfvb 6195 . . . . . . . . 9 ((1st Fn V ∧ 𝑦 ∈ V) → ((1st𝑦) = 𝑥𝑦1st 𝑥))
1513, 2, 14mp2an 707 . . . . . . . 8 ((1st𝑦) = 𝑥𝑦1st 𝑥)
16 dfid2 4994 . . . . . . . . . 10 I = {⟨𝑧, 𝑧⟩ ∣ 𝑧 = 𝑧}
1716eleq2i 2690 . . . . . . . . 9 (𝑦 ∈ I ↔ 𝑦 ∈ {⟨𝑧, 𝑧⟩ ∣ 𝑧 = 𝑧})
18 nfe1 2024 . . . . . . . . . . 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 4945 . . . . . . . . . 10 (𝑦 ∈ {⟨𝑧, 𝑧⟩ ∣ 𝑧 = 𝑧} ↔ ∃𝑧𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧))
21 equid 1936 . . . . . . . . . . . 12 𝑧 = 𝑧
2221biantru 526 . . . . . . . . . . 11 (𝑦 = ⟨𝑧, 𝑧⟩ ↔ (𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧))
2322exbii 1771 . . . . . . . . . 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 4380 . . . . . . . 8 (𝑧 = 𝑥 → ⟨𝑧, 𝑧⟩ = ⟨𝑥, 𝑥⟩)
3029eqeq2d 2631 . . . . . . 7 (𝑧 = 𝑥 → (𝑦 = ⟨𝑧, 𝑧⟩ ↔ 𝑦 = ⟨𝑥, 𝑥⟩))
311, 30ceqsexv 3228 . . . . . 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 4681 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑥(1st ↾ I )𝑦} = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = ⟨𝑥, 𝑥⟩}
36 relcnv 5464 . . 3 Rel (1st ↾ I )
37 dfrel4v 5545 . . 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 4713 . 2 (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = ⟨𝑥, 𝑥⟩}
4035, 38, 393eqtr4i 2653 1 (1st ↾ I ) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  Vcvv 3186  cop 4156   class class class wbr 4615  {copab 4674  cmpt 4675   I cid 4986  ccnv 5075  cres 5078  Rel wrel 5081   Fn wfn 5844  ontowfo 5847  cfv 5849  1st c1st 7114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-fo 5855  df-fv 5857  df-1st 7116
This theorem is referenced by: (None)
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