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Theorem fsplit 7805
 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 7804 in order to build compound functions such as (𝑥 ∈ (0[,)+∞) ↦ ((√‘𝑥) + (sin‘𝑥))). (Contributed by NM, 17-Sep-2007.) Replace use of dfid2 5431 with df-id 5428. (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 3444 . . . . 5 𝑥 ∈ V
2 vex 3444 . . . . 5 𝑦 ∈ V
31, 2brcnv 5720 . . . 4 (𝑥(1st ↾ I )𝑦𝑦(1st ↾ I )𝑥)
41brresi 5830 . . . 4 (𝑦(1st ↾ I )𝑥 ↔ (𝑦 ∈ I ∧ 𝑦1st 𝑥))
5 19.42v 1954 . . . . . 6 (∃𝑧((1st𝑦) = 𝑥𝑦 = ⟨𝑧, 𝑧⟩) ↔ ((1st𝑦) = 𝑥 ∧ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩))
6 vex 3444 . . . . . . . . . 10 𝑧 ∈ V
76, 6op1std 7691 . . . . . . . . 9 (𝑦 = ⟨𝑧, 𝑧⟩ → (1st𝑦) = 𝑧)
87eqeq1d 2800 . . . . . . . 8 (𝑦 = ⟨𝑧, 𝑧⟩ → ((1st𝑦) = 𝑥𝑧 = 𝑥))
98pm5.32ri 579 . . . . . . 7 (((1st𝑦) = 𝑥𝑦 = ⟨𝑧, 𝑧⟩) ↔ (𝑧 = 𝑥𝑦 = ⟨𝑧, 𝑧⟩))
109exbii 1849 . . . . . 6 (∃𝑧((1st𝑦) = 𝑥𝑦 = ⟨𝑧, 𝑧⟩) ↔ ∃𝑧(𝑧 = 𝑥𝑦 = ⟨𝑧, 𝑧⟩))
11 fo1st 7701 . . . . . . . . 9 1st :V–onto→V
12 fofn 6572 . . . . . . . . 9 (1st :V–onto→V → 1st Fn V)
1311, 12ax-mp 5 . . . . . . . 8 1st Fn V
14 fnbrfvb 6700 . . . . . . . 8 ((1st Fn V ∧ 𝑦 ∈ V) → ((1st𝑦) = 𝑥𝑦1st 𝑥))
1513, 2, 14mp2an 691 . . . . . . 7 ((1st𝑦) = 𝑥𝑦1st 𝑥)
16 df-id 5428 . . . . . . . . 9 I = {⟨𝑧, 𝑡⟩ ∣ 𝑧 = 𝑡}
1716eleq2i 2881 . . . . . . . 8 (𝑦 ∈ I ↔ 𝑦 ∈ {⟨𝑧, 𝑡⟩ ∣ 𝑧 = 𝑡})
18 elopab 5382 . . . . . . . 8 (𝑦 ∈ {⟨𝑧, 𝑡⟩ ∣ 𝑧 = 𝑡} ↔ ∃𝑧𝑡(𝑦 = ⟨𝑧, 𝑡⟩ ∧ 𝑧 = 𝑡))
19 ancom 464 . . . . . . . . . . . 12 ((𝑦 = ⟨𝑧, 𝑡⟩ ∧ 𝑧 = 𝑡) ↔ (𝑧 = 𝑡𝑦 = ⟨𝑧, 𝑡⟩))
20 equcom 2025 . . . . . . . . . . . . 13 (𝑧 = 𝑡𝑡 = 𝑧)
2120anbi1i 626 . . . . . . . . . . . 12 ((𝑧 = 𝑡𝑦 = ⟨𝑧, 𝑡⟩) ↔ (𝑡 = 𝑧𝑦 = ⟨𝑧, 𝑡⟩))
22 opeq2 4768 . . . . . . . . . . . . . 14 (𝑡 = 𝑧 → ⟨𝑧, 𝑡⟩ = ⟨𝑧, 𝑧⟩)
2322eqeq2d 2809 . . . . . . . . . . . . 13 (𝑡 = 𝑧 → (𝑦 = ⟨𝑧, 𝑡⟩ ↔ 𝑦 = ⟨𝑧, 𝑧⟩))
2423pm5.32i 578 . . . . . . . . . . . 12 ((𝑡 = 𝑧𝑦 = ⟨𝑧, 𝑡⟩) ↔ (𝑡 = 𝑧𝑦 = ⟨𝑧, 𝑧⟩))
2519, 21, 243bitri 300 . . . . . . . . . . 11 ((𝑦 = ⟨𝑧, 𝑡⟩ ∧ 𝑧 = 𝑡) ↔ (𝑡 = 𝑧𝑦 = ⟨𝑧, 𝑧⟩))
2625exbii 1849 . . . . . . . . . 10 (∃𝑡(𝑦 = ⟨𝑧, 𝑡⟩ ∧ 𝑧 = 𝑡) ↔ ∃𝑡(𝑡 = 𝑧𝑦 = ⟨𝑧, 𝑧⟩))
27 biidd 265 . . . . . . . . . . 11 (𝑡 = 𝑧 → (𝑦 = ⟨𝑧, 𝑧⟩ ↔ 𝑦 = ⟨𝑧, 𝑧⟩))
2827equsexvw 2011 . . . . . . . . . 10 (∃𝑡(𝑡 = 𝑧𝑦 = ⟨𝑧, 𝑧⟩) ↔ 𝑦 = ⟨𝑧, 𝑧⟩)
2926, 28bitri 278 . . . . . . . . 9 (∃𝑡(𝑦 = ⟨𝑧, 𝑡⟩ ∧ 𝑧 = 𝑡) ↔ 𝑦 = ⟨𝑧, 𝑧⟩)
3029exbii 1849 . . . . . . . 8 (∃𝑧𝑡(𝑦 = ⟨𝑧, 𝑡⟩ ∧ 𝑧 = 𝑡) ↔ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩)
3117, 18, 303bitrri 301 . . . . . . 7 (∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩ ↔ 𝑦 ∈ I )
3215, 31anbi12ci 630 . . . . . 6 (((1st𝑦) = 𝑥 ∧ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩) ↔ (𝑦 ∈ I ∧ 𝑦1st 𝑥))
335, 10, 323bitr3ri 305 . . . . 5 ((𝑦 ∈ I ∧ 𝑦1st 𝑥) ↔ ∃𝑧(𝑧 = 𝑥𝑦 = ⟨𝑧, 𝑧⟩))
34 id 22 . . . . . . . 8 (𝑧 = 𝑥𝑧 = 𝑥)
3534, 34opeq12d 4776 . . . . . . 7 (𝑧 = 𝑥 → ⟨𝑧, 𝑧⟩ = ⟨𝑥, 𝑥⟩)
3635eqeq2d 2809 . . . . . 6 (𝑧 = 𝑥 → (𝑦 = ⟨𝑧, 𝑧⟩ ↔ 𝑦 = ⟨𝑥, 𝑥⟩))
3736equsexvw 2011 . . . . 5 (∃𝑧(𝑧 = 𝑥𝑦 = ⟨𝑧, 𝑧⟩) ↔ 𝑦 = ⟨𝑥, 𝑥⟩)
3833, 37bitri 278 . . . 4 ((𝑦 ∈ I ∧ 𝑦1st 𝑥) ↔ 𝑦 = ⟨𝑥, 𝑥⟩)
393, 4, 383bitri 300 . . 3 (𝑥(1st ↾ I )𝑦𝑦 = ⟨𝑥, 𝑥⟩)
4039opabbii 5100 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑥(1st ↾ I )𝑦} = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = ⟨𝑥, 𝑥⟩}
41 relcnv 5937 . . 3 Rel (1st ↾ I )
42 dfrel4v 6017 . . 3 (Rel (1st ↾ I ) ↔ (1st ↾ I ) = {⟨𝑥, 𝑦⟩ ∣ 𝑥(1st ↾ I )𝑦})
4341, 42mpbi 233 . 2 (1st ↾ I ) = {⟨𝑥, 𝑦⟩ ∣ 𝑥(1st ↾ I )𝑦}
44 mptv 5138 . 2 (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = ⟨𝑥, 𝑥⟩}
4540, 43, 443eqtr4i 2831 1 (1st ↾ I ) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  Vcvv 3441  ⟨cop 4533   class class class wbr 5033  {copab 5095   ↦ cmpt 5113   I cid 5427  ◡ccnv 5521   ↾ cres 5524  Rel wrel 5527   Fn wfn 6324  –onto→wfo 6327  ‘cfv 6329  1st c1st 7679 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7451 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3722  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-nul 4246  df-if 4428  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-res 5534  df-iota 6288  df-fun 6331  df-fn 6332  df-f 6333  df-fo 6335  df-fv 6337  df-1st 7681 This theorem is referenced by:  fsplitfpar  7807
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