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Theorem fsplit 8142
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 8141 in order to build compound functions such as (𝑥 ∈ (0[,)+∞) ↦ ((√‘𝑥) + (sin‘𝑥))). (Contributed by NM, 17-Sep-2007.) Replace use of dfid2 5580 with df-id 5578. (Revised by BJ, 31-Dec-2023.)
Assertion
Ref Expression
fsplit (1st ↾ I ) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)

Proof of Theorem fsplit
Dummy variables 𝑦 𝑧 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3484 . . . . 5 𝑥 ∈ V
2 vex 3484 . . . . 5 𝑦 ∈ V
31, 2brcnv 5893 . . . 4 (𝑥(1st ↾ I )𝑦𝑦(1st ↾ I )𝑥)
41brresi 6006 . . . 4 (𝑦(1st ↾ I )𝑥 ↔ (𝑦 ∈ I ∧ 𝑦1st 𝑥))
5 19.42v 1953 . . . . . 6 (∃𝑧((1st𝑦) = 𝑥𝑦 = ⟨𝑧, 𝑧⟩) ↔ ((1st𝑦) = 𝑥 ∧ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩))
6 vex 3484 . . . . . . . . . 10 𝑧 ∈ V
76, 6op1std 8024 . . . . . . . . 9 (𝑦 = ⟨𝑧, 𝑧⟩ → (1st𝑦) = 𝑧)
87eqeq1d 2739 . . . . . . . 8 (𝑦 = ⟨𝑧, 𝑧⟩ → ((1st𝑦) = 𝑥𝑧 = 𝑥))
98pm5.32ri 575 . . . . . . 7 (((1st𝑦) = 𝑥𝑦 = ⟨𝑧, 𝑧⟩) ↔ (𝑧 = 𝑥𝑦 = ⟨𝑧, 𝑧⟩))
109exbii 1848 . . . . . 6 (∃𝑧((1st𝑦) = 𝑥𝑦 = ⟨𝑧, 𝑧⟩) ↔ ∃𝑧(𝑧 = 𝑥𝑦 = ⟨𝑧, 𝑧⟩))
11 fo1st 8034 . . . . . . . . 9 1st :V–onto→V
12 fofn 6822 . . . . . . . . 9 (1st :V–onto→V → 1st Fn V)
1311, 12ax-mp 5 . . . . . . . 8 1st Fn V
14 fnbrfvb 6959 . . . . . . . 8 ((1st Fn V ∧ 𝑦 ∈ V) → ((1st𝑦) = 𝑥𝑦1st 𝑥))
1513, 2, 14mp2an 692 . . . . . . 7 ((1st𝑦) = 𝑥𝑦1st 𝑥)
16 df-id 5578 . . . . . . . . 9 I = {⟨𝑧, 𝑡⟩ ∣ 𝑧 = 𝑡}
1716eleq2i 2833 . . . . . . . 8 (𝑦 ∈ I ↔ 𝑦 ∈ {⟨𝑧, 𝑡⟩ ∣ 𝑧 = 𝑡})
18 elopab 5532 . . . . . . . 8 (𝑦 ∈ {⟨𝑧, 𝑡⟩ ∣ 𝑧 = 𝑡} ↔ ∃𝑧𝑡(𝑦 = ⟨𝑧, 𝑡⟩ ∧ 𝑧 = 𝑡))
19 ancom 460 . . . . . . . . . . . 12 ((𝑦 = ⟨𝑧, 𝑡⟩ ∧ 𝑧 = 𝑡) ↔ (𝑧 = 𝑡𝑦 = ⟨𝑧, 𝑡⟩))
20 equcom 2017 . . . . . . . . . . . . 13 (𝑧 = 𝑡𝑡 = 𝑧)
2120anbi1i 624 . . . . . . . . . . . 12 ((𝑧 = 𝑡𝑦 = ⟨𝑧, 𝑡⟩) ↔ (𝑡 = 𝑧𝑦 = ⟨𝑧, 𝑡⟩))
22 opeq2 4874 . . . . . . . . . . . . . 14 (𝑡 = 𝑧 → ⟨𝑧, 𝑡⟩ = ⟨𝑧, 𝑧⟩)
2322eqeq2d 2748 . . . . . . . . . . . . 13 (𝑡 = 𝑧 → (𝑦 = ⟨𝑧, 𝑡⟩ ↔ 𝑦 = ⟨𝑧, 𝑧⟩))
2423pm5.32i 574 . . . . . . . . . . . 12 ((𝑡 = 𝑧𝑦 = ⟨𝑧, 𝑡⟩) ↔ (𝑡 = 𝑧𝑦 = ⟨𝑧, 𝑧⟩))
2519, 21, 243bitri 297 . . . . . . . . . . 11 ((𝑦 = ⟨𝑧, 𝑡⟩ ∧ 𝑧 = 𝑡) ↔ (𝑡 = 𝑧𝑦 = ⟨𝑧, 𝑧⟩))
2625exbii 1848 . . . . . . . . . 10 (∃𝑡(𝑦 = ⟨𝑧, 𝑡⟩ ∧ 𝑧 = 𝑡) ↔ ∃𝑡(𝑡 = 𝑧𝑦 = ⟨𝑧, 𝑧⟩))
27 biidd 262 . . . . . . . . . . 11 (𝑡 = 𝑧 → (𝑦 = ⟨𝑧, 𝑧⟩ ↔ 𝑦 = ⟨𝑧, 𝑧⟩))
2827equsexvw 2004 . . . . . . . . . 10 (∃𝑡(𝑡 = 𝑧𝑦 = ⟨𝑧, 𝑧⟩) ↔ 𝑦 = ⟨𝑧, 𝑧⟩)
2926, 28bitri 275 . . . . . . . . 9 (∃𝑡(𝑦 = ⟨𝑧, 𝑡⟩ ∧ 𝑧 = 𝑡) ↔ 𝑦 = ⟨𝑧, 𝑧⟩)
3029exbii 1848 . . . . . . . 8 (∃𝑧𝑡(𝑦 = ⟨𝑧, 𝑡⟩ ∧ 𝑧 = 𝑡) ↔ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩)
3117, 18, 303bitrri 298 . . . . . . 7 (∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩ ↔ 𝑦 ∈ I )
3215, 31anbi12ci 629 . . . . . 6 (((1st𝑦) = 𝑥 ∧ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩) ↔ (𝑦 ∈ I ∧ 𝑦1st 𝑥))
335, 10, 323bitr3ri 302 . . . . 5 ((𝑦 ∈ I ∧ 𝑦1st 𝑥) ↔ ∃𝑧(𝑧 = 𝑥𝑦 = ⟨𝑧, 𝑧⟩))
34 id 22 . . . . . . . 8 (𝑧 = 𝑥𝑧 = 𝑥)
3534, 34opeq12d 4881 . . . . . . 7 (𝑧 = 𝑥 → ⟨𝑧, 𝑧⟩ = ⟨𝑥, 𝑥⟩)
3635eqeq2d 2748 . . . . . 6 (𝑧 = 𝑥 → (𝑦 = ⟨𝑧, 𝑧⟩ ↔ 𝑦 = ⟨𝑥, 𝑥⟩))
3736equsexvw 2004 . . . . 5 (∃𝑧(𝑧 = 𝑥𝑦 = ⟨𝑧, 𝑧⟩) ↔ 𝑦 = ⟨𝑥, 𝑥⟩)
3833, 37bitri 275 . . . 4 ((𝑦 ∈ I ∧ 𝑦1st 𝑥) ↔ 𝑦 = ⟨𝑥, 𝑥⟩)
393, 4, 383bitri 297 . . 3 (𝑥(1st ↾ I )𝑦𝑦 = ⟨𝑥, 𝑥⟩)
4039opabbii 5210 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑥(1st ↾ I )𝑦} = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = ⟨𝑥, 𝑥⟩}
41 relcnv 6122 . . 3 Rel (1st ↾ I )
42 dfrel4v 6210 . . 3 (Rel (1st ↾ I ) ↔ (1st ↾ I ) = {⟨𝑥, 𝑦⟩ ∣ 𝑥(1st ↾ I )𝑦})
4341, 42mpbi 230 . 2 (1st ↾ I ) = {⟨𝑥, 𝑦⟩ ∣ 𝑥(1st ↾ I )𝑦}
44 mptv 5258 . 2 (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = ⟨𝑥, 𝑥⟩}
4540, 43, 443eqtr4i 2775 1 (1st ↾ I ) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  Vcvv 3480  cop 4632   class class class wbr 5143  {copab 5205  cmpt 5225   I cid 5577  ccnv 5684  cres 5687  Rel wrel 5690   Fn wfn 6556  ontowfo 6559  cfv 6561  1st c1st 8012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-1st 8014
This theorem is referenced by:  fsplitfpar  8143
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