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Theorem fsplit 7801
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 7800 in order to build compound functions such as (𝑥 ∈ (0[,)+∞) ↦ ((√‘𝑥) + (sin‘𝑥))). (Contributed by NM, 17-Sep-2007.) Replace use of dfid2 5456 with df-id 5453. (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 3495 . . . . 5 𝑥 ∈ V
2 vex 3495 . . . . 5 𝑦 ∈ V
31, 2brcnv 5746 . . . 4 (𝑥(1st ↾ I )𝑦𝑦(1st ↾ I )𝑥)
41brresi 5855 . . . 4 (𝑦(1st ↾ I )𝑥 ↔ (𝑦 ∈ I ∧ 𝑦1st 𝑥))
5 19.42v 1945 . . . . . 6 (∃𝑧((1st𝑦) = 𝑥𝑦 = ⟨𝑧, 𝑧⟩) ↔ ((1st𝑦) = 𝑥 ∧ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩))
6 vex 3495 . . . . . . . . . 10 𝑧 ∈ V
76, 6op1std 7688 . . . . . . . . 9 (𝑦 = ⟨𝑧, 𝑧⟩ → (1st𝑦) = 𝑧)
87eqeq1d 2820 . . . . . . . 8 (𝑦 = ⟨𝑧, 𝑧⟩ → ((1st𝑦) = 𝑥𝑧 = 𝑥))
98pm5.32ri 576 . . . . . . 7 (((1st𝑦) = 𝑥𝑦 = ⟨𝑧, 𝑧⟩) ↔ (𝑧 = 𝑥𝑦 = ⟨𝑧, 𝑧⟩))
109exbii 1839 . . . . . 6 (∃𝑧((1st𝑦) = 𝑥𝑦 = ⟨𝑧, 𝑧⟩) ↔ ∃𝑧(𝑧 = 𝑥𝑦 = ⟨𝑧, 𝑧⟩))
11 fo1st 7698 . . . . . . . . 9 1st :V–onto→V
12 fofn 6585 . . . . . . . . 9 (1st :V–onto→V → 1st Fn V)
1311, 12ax-mp 5 . . . . . . . 8 1st Fn V
14 fnbrfvb 6711 . . . . . . . 8 ((1st Fn V ∧ 𝑦 ∈ V) → ((1st𝑦) = 𝑥𝑦1st 𝑥))
1513, 2, 14mp2an 688 . . . . . . 7 ((1st𝑦) = 𝑥𝑦1st 𝑥)
16 df-id 5453 . . . . . . . . 9 I = {⟨𝑧, 𝑡⟩ ∣ 𝑧 = 𝑡}
1716eleq2i 2901 . . . . . . . 8 (𝑦 ∈ I ↔ 𝑦 ∈ {⟨𝑧, 𝑡⟩ ∣ 𝑧 = 𝑡})
18 elopab 5405 . . . . . . . 8 (𝑦 ∈ {⟨𝑧, 𝑡⟩ ∣ 𝑧 = 𝑡} ↔ ∃𝑧𝑡(𝑦 = ⟨𝑧, 𝑡⟩ ∧ 𝑧 = 𝑡))
19 ancom 461 . . . . . . . . . . . 12 ((𝑦 = ⟨𝑧, 𝑡⟩ ∧ 𝑧 = 𝑡) ↔ (𝑧 = 𝑡𝑦 = ⟨𝑧, 𝑡⟩))
20 equcom 2016 . . . . . . . . . . . . 13 (𝑧 = 𝑡𝑡 = 𝑧)
2120anbi1i 623 . . . . . . . . . . . 12 ((𝑧 = 𝑡𝑦 = ⟨𝑧, 𝑡⟩) ↔ (𝑡 = 𝑧𝑦 = ⟨𝑧, 𝑡⟩))
22 opeq2 4796 . . . . . . . . . . . . . 14 (𝑡 = 𝑧 → ⟨𝑧, 𝑡⟩ = ⟨𝑧, 𝑧⟩)
2322eqeq2d 2829 . . . . . . . . . . . . 13 (𝑡 = 𝑧 → (𝑦 = ⟨𝑧, 𝑡⟩ ↔ 𝑦 = ⟨𝑧, 𝑧⟩))
2423pm5.32i 575 . . . . . . . . . . . 12 ((𝑡 = 𝑧𝑦 = ⟨𝑧, 𝑡⟩) ↔ (𝑡 = 𝑧𝑦 = ⟨𝑧, 𝑧⟩))
2519, 21, 243bitri 298 . . . . . . . . . . 11 ((𝑦 = ⟨𝑧, 𝑡⟩ ∧ 𝑧 = 𝑡) ↔ (𝑡 = 𝑧𝑦 = ⟨𝑧, 𝑧⟩))
2625exbii 1839 . . . . . . . . . 10 (∃𝑡(𝑦 = ⟨𝑧, 𝑡⟩ ∧ 𝑧 = 𝑡) ↔ ∃𝑡(𝑡 = 𝑧𝑦 = ⟨𝑧, 𝑧⟩))
27 biidd 263 . . . . . . . . . . 11 (𝑡 = 𝑧 → (𝑦 = ⟨𝑧, 𝑧⟩ ↔ 𝑦 = ⟨𝑧, 𝑧⟩))
2827equsexvw 2002 . . . . . . . . . 10 (∃𝑡(𝑡 = 𝑧𝑦 = ⟨𝑧, 𝑧⟩) ↔ 𝑦 = ⟨𝑧, 𝑧⟩)
2926, 28bitri 276 . . . . . . . . 9 (∃𝑡(𝑦 = ⟨𝑧, 𝑡⟩ ∧ 𝑧 = 𝑡) ↔ 𝑦 = ⟨𝑧, 𝑧⟩)
3029exbii 1839 . . . . . . . 8 (∃𝑧𝑡(𝑦 = ⟨𝑧, 𝑡⟩ ∧ 𝑧 = 𝑡) ↔ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩)
3117, 18, 303bitrri 299 . . . . . . 7 (∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩ ↔ 𝑦 ∈ I )
3215, 31anbi12ci 627 . . . . . 6 (((1st𝑦) = 𝑥 ∧ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩) ↔ (𝑦 ∈ I ∧ 𝑦1st 𝑥))
335, 10, 323bitr3ri 303 . . . . 5 ((𝑦 ∈ I ∧ 𝑦1st 𝑥) ↔ ∃𝑧(𝑧 = 𝑥𝑦 = ⟨𝑧, 𝑧⟩))
34 id 22 . . . . . . . 8 (𝑧 = 𝑥𝑧 = 𝑥)
3534, 34opeq12d 4803 . . . . . . 7 (𝑧 = 𝑥 → ⟨𝑧, 𝑧⟩ = ⟨𝑥, 𝑥⟩)
3635eqeq2d 2829 . . . . . 6 (𝑧 = 𝑥 → (𝑦 = ⟨𝑧, 𝑧⟩ ↔ 𝑦 = ⟨𝑥, 𝑥⟩))
3736equsexvw 2002 . . . . 5 (∃𝑧(𝑧 = 𝑥𝑦 = ⟨𝑧, 𝑧⟩) ↔ 𝑦 = ⟨𝑥, 𝑥⟩)
3833, 37bitri 276 . . . 4 ((𝑦 ∈ I ∧ 𝑦1st 𝑥) ↔ 𝑦 = ⟨𝑥, 𝑥⟩)
393, 4, 383bitri 298 . . 3 (𝑥(1st ↾ I )𝑦𝑦 = ⟨𝑥, 𝑥⟩)
4039opabbii 5124 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑥(1st ↾ I )𝑦} = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = ⟨𝑥, 𝑥⟩}
41 relcnv 5960 . . 3 Rel (1st ↾ I )
42 dfrel4v 6040 . . 3 (Rel (1st ↾ I ) ↔ (1st ↾ I ) = {⟨𝑥, 𝑦⟩ ∣ 𝑥(1st ↾ I )𝑦})
4341, 42mpbi 231 . 2 (1st ↾ I ) = {⟨𝑥, 𝑦⟩ ∣ 𝑥(1st ↾ I )𝑦}
44 mptv 5162 . 2 (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = ⟨𝑥, 𝑥⟩}
4540, 43, 443eqtr4i 2851 1 (1st ↾ I ) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1528  wex 1771  wcel 2105  Vcvv 3492  cop 4563   class class class wbr 5057  {copab 5119  cmpt 5137   I cid 5452  ccnv 5547  cres 5550  Rel wrel 5553   Fn wfn 6343  ontowfo 6346  cfv 6348  1st c1st 7676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fo 6354  df-fv 6356  df-1st 7678
This theorem is referenced by:  fsplitfpar  7803
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