MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fsplit Structured version   Visualization version   GIF version

Theorem fsplit 8002
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 8001 in order to build compound functions such as (𝑥 ∈ (0[,)+∞) ↦ ((√‘𝑥) + (sin‘𝑥))). (Contributed by NM, 17-Sep-2007.) Replace use of dfid2 5509 with df-id 5507. (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 3445 . . . . 5 𝑥 ∈ V
2 vex 3445 . . . . 5 𝑦 ∈ V
31, 2brcnv 5811 . . . 4 (𝑥(1st ↾ I )𝑦𝑦(1st ↾ I )𝑥)
41brresi 5919 . . . 4 (𝑦(1st ↾ I )𝑥 ↔ (𝑦 ∈ I ∧ 𝑦1st 𝑥))
5 19.42v 1956 . . . . . 6 (∃𝑧((1st𝑦) = 𝑥𝑦 = ⟨𝑧, 𝑧⟩) ↔ ((1st𝑦) = 𝑥 ∧ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩))
6 vex 3445 . . . . . . . . . 10 𝑧 ∈ V
76, 6op1std 7886 . . . . . . . . 9 (𝑦 = ⟨𝑧, 𝑧⟩ → (1st𝑦) = 𝑧)
87eqeq1d 2739 . . . . . . . 8 (𝑦 = ⟨𝑧, 𝑧⟩ → ((1st𝑦) = 𝑥𝑧 = 𝑥))
98pm5.32ri 576 . . . . . . 7 (((1st𝑦) = 𝑥𝑦 = ⟨𝑧, 𝑧⟩) ↔ (𝑧 = 𝑥𝑦 = ⟨𝑧, 𝑧⟩))
109exbii 1849 . . . . . 6 (∃𝑧((1st𝑦) = 𝑥𝑦 = ⟨𝑧, 𝑧⟩) ↔ ∃𝑧(𝑧 = 𝑥𝑦 = ⟨𝑧, 𝑧⟩))
11 fo1st 7896 . . . . . . . . 9 1st :V–onto→V
12 fofn 6727 . . . . . . . . 9 (1st :V–onto→V → 1st Fn V)
1311, 12ax-mp 5 . . . . . . . 8 1st Fn V
14 fnbrfvb 6861 . . . . . . . 8 ((1st Fn V ∧ 𝑦 ∈ V) → ((1st𝑦) = 𝑥𝑦1st 𝑥))
1513, 2, 14mp2an 689 . . . . . . 7 ((1st𝑦) = 𝑥𝑦1st 𝑥)
16 df-id 5507 . . . . . . . . 9 I = {⟨𝑧, 𝑡⟩ ∣ 𝑧 = 𝑡}
1716eleq2i 2829 . . . . . . . 8 (𝑦 ∈ I ↔ 𝑦 ∈ {⟨𝑧, 𝑡⟩ ∣ 𝑧 = 𝑡})
18 elopab 5460 . . . . . . . 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 2020 . . . . . . . . . . . . 13 (𝑧 = 𝑡𝑡 = 𝑧)
2120anbi1i 624 . . . . . . . . . . . 12 ((𝑧 = 𝑡𝑦 = ⟨𝑧, 𝑡⟩) ↔ (𝑡 = 𝑧𝑦 = ⟨𝑧, 𝑡⟩))
22 opeq2 4816 . . . . . . . . . . . . . 14 (𝑡 = 𝑧 → ⟨𝑧, 𝑡⟩ = ⟨𝑧, 𝑧⟩)
2322eqeq2d 2748 . . . . . . . . . . . . 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 1849 . . . . . . . . . 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 2007 . . . . . . . . . 10 (∃𝑡(𝑡 = 𝑧𝑦 = ⟨𝑧, 𝑧⟩) ↔ 𝑦 = ⟨𝑧, 𝑧⟩)
2926, 28bitri 274 . . . . . . . . 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 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 4823 . . . . . . 7 (𝑧 = 𝑥 → ⟨𝑧, 𝑧⟩ = ⟨𝑥, 𝑥⟩)
3635eqeq2d 2748 . . . . . 6 (𝑧 = 𝑥 → (𝑦 = ⟨𝑧, 𝑧⟩ ↔ 𝑦 = ⟨𝑥, 𝑥⟩))
3736equsexvw 2007 . . . . 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 5154 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑥(1st ↾ I )𝑦} = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = ⟨𝑥, 𝑥⟩}
41 relcnv 6029 . . 3 Rel (1st ↾ I )
42 dfrel4v 6115 . . 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 5203 . 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 205  wa 396   = wceq 1540  wex 1780  wcel 2105  Vcvv 3441  cop 4577   class class class wbr 5087  {copab 5149  cmpt 5170   I cid 5506  ccnv 5606  cres 5609  Rel wrel 5612   Fn wfn 6460  ontowfo 6463  cfv 6465  1st c1st 7874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-sep 5238  ax-nul 5245  ax-pr 5367  ax-un 7628
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3443  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4268  df-if 4472  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4851  df-br 5088  df-opab 5150  df-mpt 5171  df-id 5507  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-fo 6471  df-fv 6473  df-1st 7876
This theorem is referenced by:  fsplitfpar  8003
  Copyright terms: Public domain W3C validator