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Theorem fsplit 7433
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 7432 in order to build compound functions such as 𝑦 = ((√‘𝑥) + (abs‘𝑥)). (Contributed by NM, 17-Sep-2007.)
Assertion
Ref Expression
fsplit (1st ↾ I ) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)

Proof of Theorem fsplit
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3354 . . . . 5 𝑥 ∈ V
2 vex 3354 . . . . 5 𝑦 ∈ V
31, 2brcnv 5443 . . . 4 (𝑥(1st ↾ I )𝑦𝑦(1st ↾ I )𝑥)
41brres 5543 . . . . 5 (𝑦(1st ↾ I )𝑥 ↔ (𝑦1st 𝑥𝑦 ∈ I ))
5 19.42v 2033 . . . . . . 7 (∃𝑧((1st𝑦) = 𝑥𝑦 = ⟨𝑧, 𝑧⟩) ↔ ((1st𝑦) = 𝑥 ∧ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩))
6 vex 3354 . . . . . . . . . . 11 𝑧 ∈ V
76, 6op1std 7325 . . . . . . . . . 10 (𝑦 = ⟨𝑧, 𝑧⟩ → (1st𝑦) = 𝑧)
87eqeq1d 2773 . . . . . . . . 9 (𝑦 = ⟨𝑧, 𝑧⟩ → ((1st𝑦) = 𝑥𝑧 = 𝑥))
98pm5.32ri 565 . . . . . . . 8 (((1st𝑦) = 𝑥𝑦 = ⟨𝑧, 𝑧⟩) ↔ (𝑧 = 𝑥𝑦 = ⟨𝑧, 𝑧⟩))
109exbii 1924 . . . . . . 7 (∃𝑧((1st𝑦) = 𝑥𝑦 = ⟨𝑧, 𝑧⟩) ↔ ∃𝑧(𝑧 = 𝑥𝑦 = ⟨𝑧, 𝑧⟩))
11 fo1st 7335 . . . . . . . . . 10 1st :V–onto→V
12 fofn 6258 . . . . . . . . . 10 (1st :V–onto→V → 1st Fn V)
1311, 12ax-mp 5 . . . . . . . . 9 1st Fn V
14 fnbrfvb 6377 . . . . . . . . 9 ((1st Fn V ∧ 𝑦 ∈ V) → ((1st𝑦) = 𝑥𝑦1st 𝑥))
1513, 2, 14mp2an 672 . . . . . . . 8 ((1st𝑦) = 𝑥𝑦1st 𝑥)
16 dfid2 5160 . . . . . . . . . 10 I = {⟨𝑧, 𝑧⟩ ∣ 𝑧 = 𝑧}
1716eleq2i 2842 . . . . . . . . 9 (𝑦 ∈ I ↔ 𝑦 ∈ {⟨𝑧, 𝑧⟩ ∣ 𝑧 = 𝑧})
18 nfe1 2183 . . . . . . . . . . 11 𝑧𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧)
191819.9 2228 . . . . . . . . . 10 (∃𝑧𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧) ↔ ∃𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧))
20 elopab 5116 . . . . . . . . . 10 (𝑦 ∈ {⟨𝑧, 𝑧⟩ ∣ 𝑧 = 𝑧} ↔ ∃𝑧𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧))
21 equid 2097 . . . . . . . . . . . 12 𝑧 = 𝑧
2221biantru 519 . . . . . . . . . . 11 (𝑦 = ⟨𝑧, 𝑧⟩ ↔ (𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧))
2322exbii 1924 . . . . . . . . . 10 (∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩ ↔ ∃𝑧(𝑦 = ⟨𝑧, 𝑧⟩ ∧ 𝑧 = 𝑧))
2419, 20, 233bitr4i 292 . . . . . . . . 9 (𝑦 ∈ {⟨𝑧, 𝑧⟩ ∣ 𝑧 = 𝑧} ↔ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩)
2517, 24bitr2i 265 . . . . . . . 8 (∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩ ↔ 𝑦 ∈ I )
2615, 25anbi12i 612 . . . . . . 7 (((1st𝑦) = 𝑥 ∧ ∃𝑧 𝑦 = ⟨𝑧, 𝑧⟩) ↔ (𝑦1st 𝑥𝑦 ∈ I ))
275, 10, 263bitr3ri 291 . . . . . 6 ((𝑦1st 𝑥𝑦 ∈ I ) ↔ ∃𝑧(𝑧 = 𝑥𝑦 = ⟨𝑧, 𝑧⟩))
28 id 22 . . . . . . . . 9 (𝑧 = 𝑥𝑧 = 𝑥)
2928, 28opeq12d 4547 . . . . . . . 8 (𝑧 = 𝑥 → ⟨𝑧, 𝑧⟩ = ⟨𝑥, 𝑥⟩)
3029eqeq2d 2781 . . . . . . 7 (𝑧 = 𝑥 → (𝑦 = ⟨𝑧, 𝑧⟩ ↔ 𝑦 = ⟨𝑥, 𝑥⟩))
311, 30ceqsexv 3394 . . . . . 6 (∃𝑧(𝑧 = 𝑥𝑦 = ⟨𝑧, 𝑧⟩) ↔ 𝑦 = ⟨𝑥, 𝑥⟩)
3227, 31bitri 264 . . . . 5 ((𝑦1st 𝑥𝑦 ∈ I ) ↔ 𝑦 = ⟨𝑥, 𝑥⟩)
334, 32bitri 264 . . . 4 (𝑦(1st ↾ I )𝑥𝑦 = ⟨𝑥, 𝑥⟩)
343, 33bitri 264 . . 3 (𝑥(1st ↾ I )𝑦𝑦 = ⟨𝑥, 𝑥⟩)
3534opabbii 4851 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑥(1st ↾ I )𝑦} = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = ⟨𝑥, 𝑥⟩}
36 relcnv 5644 . . 3 Rel (1st ↾ I )
37 dfrel4v 5725 . . 3 (Rel (1st ↾ I ) ↔ (1st ↾ I ) = {⟨𝑥, 𝑦⟩ ∣ 𝑥(1st ↾ I )𝑦})
3836, 37mpbi 220 . 2 (1st ↾ I ) = {⟨𝑥, 𝑦⟩ ∣ 𝑥(1st ↾ I )𝑦}
39 mptv 4885 . 2 (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = ⟨𝑥, 𝑥⟩}
4035, 38, 393eqtr4i 2803 1 (1st ↾ I ) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382   = wceq 1631  wex 1852  wcel 2145  Vcvv 3351  cop 4322   class class class wbr 4786  {copab 4846  cmpt 4863   I cid 5156  ccnv 5248  cres 5251  Rel wrel 5254   Fn wfn 6026  ontowfo 6029  cfv 6031  1st c1st 7313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fo 6037  df-fv 6039  df-1st 7315
This theorem is referenced by: (None)
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