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Theorem fsplit 8049
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 8048 in order to build compound functions such as (𝑥 ∈ (0[,)+∞) ↦ ((√‘𝑥) + (sin‘𝑥))). (Contributed by NM, 17-Sep-2007.) Replace use of dfid2 5533 with df-id 5531. (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 3449 . . . . 5 𝑥 ∈ V
2 vex 3449 . . . . 5 𝑦 ∈ V
31, 2brcnv 5838 . . . 4 (𝑥(1st ↾ I )𝑦𝑦(1st ↾ I )𝑥)
41brresi 5946 . . . 4 (𝑦(1st ↾ I )𝑥 ↔ (𝑦 ∈ I ∧ 𝑦1st 𝑥))
5 19.42v 1957 . . . . . 6 (∃𝑧((1st𝑦) = 𝑥𝑦 = ⟨𝑧, 𝑧⟩) ↔ ((1st𝑦) = 𝑥 ∧ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩))
6 vex 3449 . . . . . . . . . 10 𝑧 ∈ V
76, 6op1std 7931 . . . . . . . . 9 (𝑦 = ⟨𝑧, 𝑧⟩ → (1st𝑦) = 𝑧)
87eqeq1d 2738 . . . . . . . 8 (𝑦 = ⟨𝑧, 𝑧⟩ → ((1st𝑦) = 𝑥𝑧 = 𝑥))
98pm5.32ri 576 . . . . . . 7 (((1st𝑦) = 𝑥𝑦 = ⟨𝑧, 𝑧⟩) ↔ (𝑧 = 𝑥𝑦 = ⟨𝑧, 𝑧⟩))
109exbii 1850 . . . . . 6 (∃𝑧((1st𝑦) = 𝑥𝑦 = ⟨𝑧, 𝑧⟩) ↔ ∃𝑧(𝑧 = 𝑥𝑦 = ⟨𝑧, 𝑧⟩))
11 fo1st 7941 . . . . . . . . 9 1st :V–onto→V
12 fofn 6758 . . . . . . . . 9 (1st :V–onto→V → 1st Fn V)
1311, 12ax-mp 5 . . . . . . . 8 1st Fn V
14 fnbrfvb 6895 . . . . . . . 8 ((1st Fn V ∧ 𝑦 ∈ V) → ((1st𝑦) = 𝑥𝑦1st 𝑥))
1513, 2, 14mp2an 690 . . . . . . 7 ((1st𝑦) = 𝑥𝑦1st 𝑥)
16 df-id 5531 . . . . . . . . 9 I = {⟨𝑧, 𝑡⟩ ∣ 𝑧 = 𝑡}
1716eleq2i 2829 . . . . . . . 8 (𝑦 ∈ I ↔ 𝑦 ∈ {⟨𝑧, 𝑡⟩ ∣ 𝑧 = 𝑡})
18 elopab 5484 . . . . . . . 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 2021 . . . . . . . . . . . . 13 (𝑧 = 𝑡𝑡 = 𝑧)
2120anbi1i 624 . . . . . . . . . . . 12 ((𝑧 = 𝑡𝑦 = ⟨𝑧, 𝑡⟩) ↔ (𝑡 = 𝑧𝑦 = ⟨𝑧, 𝑡⟩))
22 opeq2 4831 . . . . . . . . . . . . . 14 (𝑡 = 𝑧 → ⟨𝑧, 𝑡⟩ = ⟨𝑧, 𝑧⟩)
2322eqeq2d 2747 . . . . . . . . . . . . 13 (𝑡 = 𝑧 → (𝑦 = ⟨𝑧, 𝑡⟩ ↔ 𝑦 = ⟨𝑧, 𝑧⟩))
2423pm5.32i 575 . . . . . . . . . . . 12 ((𝑡 = 𝑧𝑦 = ⟨𝑧, 𝑡⟩) ↔ (𝑡 = 𝑧𝑦 = ⟨𝑧, 𝑧⟩))
2519, 21, 243bitri 296 . . . . . . . . . . 11 ((𝑦 = ⟨𝑧, 𝑡⟩ ∧ 𝑧 = 𝑡) ↔ (𝑡 = 𝑧𝑦 = ⟨𝑧, 𝑧⟩))
2625exbii 1850 . . . . . . . . . 10 (∃𝑡(𝑦 = ⟨𝑧, 𝑡⟩ ∧ 𝑧 = 𝑡) ↔ ∃𝑡(𝑡 = 𝑧𝑦 = ⟨𝑧, 𝑧⟩))
27 biidd 261 . . . . . . . . . . 11 (𝑡 = 𝑧 → (𝑦 = ⟨𝑧, 𝑧⟩ ↔ 𝑦 = ⟨𝑧, 𝑧⟩))
2827equsexvw 2008 . . . . . . . . . 10 (∃𝑡(𝑡 = 𝑧𝑦 = ⟨𝑧, 𝑧⟩) ↔ 𝑦 = ⟨𝑧, 𝑧⟩)
2926, 28bitri 274 . . . . . . . . 9 (∃𝑡(𝑦 = ⟨𝑧, 𝑡⟩ ∧ 𝑧 = 𝑡) ↔ 𝑦 = ⟨𝑧, 𝑧⟩)
3029exbii 1850 . . . . . . . 8 (∃𝑧𝑡(𝑦 = ⟨𝑧, 𝑡⟩ ∧ 𝑧 = 𝑡) ↔ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩)
3117, 18, 303bitrri 297 . . . . . . 7 (∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩ ↔ 𝑦 ∈ I )
3215, 31anbi12ci 628 . . . . . 6 (((1st𝑦) = 𝑥 ∧ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩) ↔ (𝑦 ∈ I ∧ 𝑦1st 𝑥))
335, 10, 323bitr3ri 301 . . . . 5 ((𝑦 ∈ I ∧ 𝑦1st 𝑥) ↔ ∃𝑧(𝑧 = 𝑥𝑦 = ⟨𝑧, 𝑧⟩))
34 id 22 . . . . . . . 8 (𝑧 = 𝑥𝑧 = 𝑥)
3534, 34opeq12d 4838 . . . . . . 7 (𝑧 = 𝑥 → ⟨𝑧, 𝑧⟩ = ⟨𝑥, 𝑥⟩)
3635eqeq2d 2747 . . . . . 6 (𝑧 = 𝑥 → (𝑦 = ⟨𝑧, 𝑧⟩ ↔ 𝑦 = ⟨𝑥, 𝑥⟩))
3736equsexvw 2008 . . . . 5 (∃𝑧(𝑧 = 𝑥𝑦 = ⟨𝑧, 𝑧⟩) ↔ 𝑦 = ⟨𝑥, 𝑥⟩)
3833, 37bitri 274 . . . 4 ((𝑦 ∈ I ∧ 𝑦1st 𝑥) ↔ 𝑦 = ⟨𝑥, 𝑥⟩)
393, 4, 383bitri 296 . . 3 (𝑥(1st ↾ I )𝑦𝑦 = ⟨𝑥, 𝑥⟩)
4039opabbii 5172 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑥(1st ↾ I )𝑦} = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = ⟨𝑥, 𝑥⟩}
41 relcnv 6056 . . 3 Rel (1st ↾ I )
42 dfrel4v 6142 . . 3 (Rel (1st ↾ I ) ↔ (1st ↾ I ) = {⟨𝑥, 𝑦⟩ ∣ 𝑥(1st ↾ I )𝑦})
4341, 42mpbi 229 . 2 (1st ↾ I ) = {⟨𝑥, 𝑦⟩ ∣ 𝑥(1st ↾ I )𝑦}
44 mptv 5221 . 2 (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = ⟨𝑥, 𝑥⟩}
4540, 43, 443eqtr4i 2774 1 (1st ↾ I ) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1541  wex 1781  wcel 2106  Vcvv 3445  cop 4592   class class class wbr 5105  {copab 5167  cmpt 5188   I cid 5530  ccnv 5632  cres 5635  Rel wrel 5638   Fn wfn 6491  ontowfo 6494  cfv 6496  1st c1st 7919
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fo 6502  df-fv 6504  df-1st 7921
This theorem is referenced by:  fsplitfpar  8050
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