MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fsplit Structured version   Visualization version   GIF version

Theorem fsplit 8103
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 8102 in order to build compound functions such as (π‘₯ ∈ (0[,)+∞) ↦ ((βˆšβ€˜π‘₯) + (sinβ€˜π‘₯))). (Contributed by NM, 17-Sep-2007.) Replace use of dfid2 5577 with df-id 5575. (Revised by BJ, 31-Dec-2023.)
Assertion
Ref Expression
fsplit β—‘(1st β†Ύ I ) = (π‘₯ ∈ V ↦ ⟨π‘₯, π‘₯⟩)

Proof of Theorem fsplit
Dummy variables 𝑦 𝑧 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3479 . . . . 5 π‘₯ ∈ V
2 vex 3479 . . . . 5 𝑦 ∈ V
31, 2brcnv 5883 . . . 4 (π‘₯β—‘(1st β†Ύ I )𝑦 ↔ 𝑦(1st β†Ύ I )π‘₯)
41brresi 5991 . . . 4 (𝑦(1st β†Ύ I )π‘₯ ↔ (𝑦 ∈ I ∧ 𝑦1st π‘₯))
5 19.42v 1958 . . . . . 6 (βˆƒπ‘§((1st β€˜π‘¦) = π‘₯ ∧ 𝑦 = βŸ¨π‘§, π‘§βŸ©) ↔ ((1st β€˜π‘¦) = π‘₯ ∧ βˆƒπ‘§ 𝑦 = βŸ¨π‘§, π‘§βŸ©))
6 vex 3479 . . . . . . . . . 10 𝑧 ∈ V
76, 6op1std 7985 . . . . . . . . 9 (𝑦 = βŸ¨π‘§, π‘§βŸ© β†’ (1st β€˜π‘¦) = 𝑧)
87eqeq1d 2735 . . . . . . . 8 (𝑦 = βŸ¨π‘§, π‘§βŸ© β†’ ((1st β€˜π‘¦) = π‘₯ ↔ 𝑧 = π‘₯))
98pm5.32ri 577 . . . . . . 7 (((1st β€˜π‘¦) = π‘₯ ∧ 𝑦 = βŸ¨π‘§, π‘§βŸ©) ↔ (𝑧 = π‘₯ ∧ 𝑦 = βŸ¨π‘§, π‘§βŸ©))
109exbii 1851 . . . . . 6 (βˆƒπ‘§((1st β€˜π‘¦) = π‘₯ ∧ 𝑦 = βŸ¨π‘§, π‘§βŸ©) ↔ βˆƒπ‘§(𝑧 = π‘₯ ∧ 𝑦 = βŸ¨π‘§, π‘§βŸ©))
11 fo1st 7995 . . . . . . . . 9 1st :V–ontoβ†’V
12 fofn 6808 . . . . . . . . 9 (1st :V–ontoβ†’V β†’ 1st Fn V)
1311, 12ax-mp 5 . . . . . . . 8 1st Fn V
14 fnbrfvb 6945 . . . . . . . 8 ((1st Fn V ∧ 𝑦 ∈ V) β†’ ((1st β€˜π‘¦) = π‘₯ ↔ 𝑦1st π‘₯))
1513, 2, 14mp2an 691 . . . . . . 7 ((1st β€˜π‘¦) = π‘₯ ↔ 𝑦1st π‘₯)
16 df-id 5575 . . . . . . . . 9 I = {βŸ¨π‘§, π‘‘βŸ© ∣ 𝑧 = 𝑑}
1716eleq2i 2826 . . . . . . . 8 (𝑦 ∈ I ↔ 𝑦 ∈ {βŸ¨π‘§, π‘‘βŸ© ∣ 𝑧 = 𝑑})
18 elopab 5528 . . . . . . . 8 (𝑦 ∈ {βŸ¨π‘§, π‘‘βŸ© ∣ 𝑧 = 𝑑} ↔ βˆƒπ‘§βˆƒπ‘‘(𝑦 = βŸ¨π‘§, π‘‘βŸ© ∧ 𝑧 = 𝑑))
19 ancom 462 . . . . . . . . . . . 12 ((𝑦 = βŸ¨π‘§, π‘‘βŸ© ∧ 𝑧 = 𝑑) ↔ (𝑧 = 𝑑 ∧ 𝑦 = βŸ¨π‘§, π‘‘βŸ©))
20 equcom 2022 . . . . . . . . . . . . 13 (𝑧 = 𝑑 ↔ 𝑑 = 𝑧)
2120anbi1i 625 . . . . . . . . . . . 12 ((𝑧 = 𝑑 ∧ 𝑦 = βŸ¨π‘§, π‘‘βŸ©) ↔ (𝑑 = 𝑧 ∧ 𝑦 = βŸ¨π‘§, π‘‘βŸ©))
22 opeq2 4875 . . . . . . . . . . . . . 14 (𝑑 = 𝑧 β†’ βŸ¨π‘§, π‘‘βŸ© = βŸ¨π‘§, π‘§βŸ©)
2322eqeq2d 2744 . . . . . . . . . . . . 13 (𝑑 = 𝑧 β†’ (𝑦 = βŸ¨π‘§, π‘‘βŸ© ↔ 𝑦 = βŸ¨π‘§, π‘§βŸ©))
2423pm5.32i 576 . . . . . . . . . . . 12 ((𝑑 = 𝑧 ∧ 𝑦 = βŸ¨π‘§, π‘‘βŸ©) ↔ (𝑑 = 𝑧 ∧ 𝑦 = βŸ¨π‘§, π‘§βŸ©))
2519, 21, 243bitri 297 . . . . . . . . . . 11 ((𝑦 = βŸ¨π‘§, π‘‘βŸ© ∧ 𝑧 = 𝑑) ↔ (𝑑 = 𝑧 ∧ 𝑦 = βŸ¨π‘§, π‘§βŸ©))
2625exbii 1851 . . . . . . . . . 10 (βˆƒπ‘‘(𝑦 = βŸ¨π‘§, π‘‘βŸ© ∧ 𝑧 = 𝑑) ↔ βˆƒπ‘‘(𝑑 = 𝑧 ∧ 𝑦 = βŸ¨π‘§, π‘§βŸ©))
27 biidd 262 . . . . . . . . . . 11 (𝑑 = 𝑧 β†’ (𝑦 = βŸ¨π‘§, π‘§βŸ© ↔ 𝑦 = βŸ¨π‘§, π‘§βŸ©))
2827equsexvw 2009 . . . . . . . . . 10 (βˆƒπ‘‘(𝑑 = 𝑧 ∧ 𝑦 = βŸ¨π‘§, π‘§βŸ©) ↔ 𝑦 = βŸ¨π‘§, π‘§βŸ©)
2926, 28bitri 275 . . . . . . . . 9 (βˆƒπ‘‘(𝑦 = βŸ¨π‘§, π‘‘βŸ© ∧ 𝑧 = 𝑑) ↔ 𝑦 = βŸ¨π‘§, π‘§βŸ©)
3029exbii 1851 . . . . . . . 8 (βˆƒπ‘§βˆƒπ‘‘(𝑦 = βŸ¨π‘§, π‘‘βŸ© ∧ 𝑧 = 𝑑) ↔ βˆƒπ‘§ 𝑦 = βŸ¨π‘§, π‘§βŸ©)
3117, 18, 303bitrri 298 . . . . . . 7 (βˆƒπ‘§ 𝑦 = βŸ¨π‘§, π‘§βŸ© ↔ 𝑦 ∈ I )
3215, 31anbi12ci 629 . . . . . 6 (((1st β€˜π‘¦) = π‘₯ ∧ βˆƒπ‘§ 𝑦 = βŸ¨π‘§, π‘§βŸ©) ↔ (𝑦 ∈ I ∧ 𝑦1st π‘₯))
335, 10, 323bitr3ri 302 . . . . 5 ((𝑦 ∈ I ∧ 𝑦1st π‘₯) ↔ βˆƒπ‘§(𝑧 = π‘₯ ∧ 𝑦 = βŸ¨π‘§, π‘§βŸ©))
34 id 22 . . . . . . . 8 (𝑧 = π‘₯ β†’ 𝑧 = π‘₯)
3534, 34opeq12d 4882 . . . . . . 7 (𝑧 = π‘₯ β†’ βŸ¨π‘§, π‘§βŸ© = ⟨π‘₯, π‘₯⟩)
3635eqeq2d 2744 . . . . . 6 (𝑧 = π‘₯ β†’ (𝑦 = βŸ¨π‘§, π‘§βŸ© ↔ 𝑦 = ⟨π‘₯, π‘₯⟩))
3736equsexvw 2009 . . . . 5 (βˆƒπ‘§(𝑧 = π‘₯ ∧ 𝑦 = βŸ¨π‘§, π‘§βŸ©) ↔ 𝑦 = ⟨π‘₯, π‘₯⟩)
3833, 37bitri 275 . . . 4 ((𝑦 ∈ I ∧ 𝑦1st π‘₯) ↔ 𝑦 = ⟨π‘₯, π‘₯⟩)
393, 4, 383bitri 297 . . 3 (π‘₯β—‘(1st β†Ύ I )𝑦 ↔ 𝑦 = ⟨π‘₯, π‘₯⟩)
4039opabbii 5216 . 2 {⟨π‘₯, π‘¦βŸ© ∣ π‘₯β—‘(1st β†Ύ I )𝑦} = {⟨π‘₯, π‘¦βŸ© ∣ 𝑦 = ⟨π‘₯, π‘₯⟩}
41 relcnv 6104 . . 3 Rel β—‘(1st β†Ύ I )
42 dfrel4v 6190 . . 3 (Rel β—‘(1st β†Ύ I ) ↔ β—‘(1st β†Ύ I ) = {⟨π‘₯, π‘¦βŸ© ∣ π‘₯β—‘(1st β†Ύ I )𝑦})
4341, 42mpbi 229 . 2 β—‘(1st β†Ύ I ) = {⟨π‘₯, π‘¦βŸ© ∣ π‘₯β—‘(1st β†Ύ I )𝑦}
44 mptv 5265 . 2 (π‘₯ ∈ V ↦ ⟨π‘₯, π‘₯⟩) = {⟨π‘₯, π‘¦βŸ© ∣ 𝑦 = ⟨π‘₯, π‘₯⟩}
4540, 43, 443eqtr4i 2771 1 β—‘(1st β†Ύ I ) = (π‘₯ ∈ V ↦ ⟨π‘₯, π‘₯⟩)
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  Vcvv 3475  βŸ¨cop 4635   class class class wbr 5149  {copab 5211   ↦ cmpt 5232   I cid 5574  β—‘ccnv 5676   β†Ύ cres 5679  Rel wrel 5682   Fn wfn 6539  β€“ontoβ†’wfo 6542  β€˜cfv 6544  1st c1st 7973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-1st 7975
This theorem is referenced by:  fsplitfpar  8104
  Copyright terms: Public domain W3C validator