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Theorem fsplit 8050
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 8049 in order to build compound functions such as (π‘₯ ∈ (0[,)+∞) ↦ ((βˆšβ€˜π‘₯) + (sinβ€˜π‘₯))). (Contributed by NM, 17-Sep-2007.) Replace use of dfid2 5534 with df-id 5532. (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 3448 . . . . 5 π‘₯ ∈ V
2 vex 3448 . . . . 5 𝑦 ∈ V
31, 2brcnv 5839 . . . 4 (π‘₯β—‘(1st β†Ύ I )𝑦 ↔ 𝑦(1st β†Ύ I )π‘₯)
41brresi 5947 . . . 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 3448 . . . . . . . . . 10 𝑧 ∈ V
76, 6op1std 7932 . . . . . . . . 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 7942 . . . . . . . . 9 1st :V–ontoβ†’V
12 fofn 6759 . . . . . . . . 9 (1st :V–ontoβ†’V β†’ 1st Fn V)
1311, 12ax-mp 5 . . . . . . . 8 1st Fn V
14 fnbrfvb 6896 . . . . . . . 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 5532 . . . . . . . . 9 I = {βŸ¨π‘§, π‘‘βŸ© ∣ 𝑧 = 𝑑}
1716eleq2i 2826 . . . . . . . 8 (𝑦 ∈ I ↔ 𝑦 ∈ {βŸ¨π‘§, π‘‘βŸ© ∣ 𝑧 = 𝑑})
18 elopab 5485 . . . . . . . 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 4832 . . . . . . . . . . . . . 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 4839 . . . . . . 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 5173 . 2 {⟨π‘₯, π‘¦βŸ© ∣ π‘₯β—‘(1st β†Ύ I )𝑦} = {⟨π‘₯, π‘¦βŸ© ∣ 𝑦 = ⟨π‘₯, π‘₯⟩}
41 relcnv 6057 . . 3 Rel β—‘(1st β†Ύ I )
42 dfrel4v 6143 . . 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 5222 . 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 3444  βŸ¨cop 4593   class class class wbr 5106  {copab 5168   ↦ cmpt 5189   I cid 5531  β—‘ccnv 5633   β†Ύ cres 5636  Rel wrel 5639   Fn wfn 6492  β€“ontoβ†’wfo 6495  β€˜cfv 6497  1st c1st 7920
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 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
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 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fo 6503  df-fv 6505  df-1st 7922
This theorem is referenced by:  fsplitfpar  8051
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