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

Theorem fnsnsplit 7205
Description: Split a function into a single point and all the rest. (Contributed by Stefan O'Rear, 27-Feb-2015.)
Assertion
Ref Expression
fnsnsplit ((𝐹 Fn 𝐴𝑋𝐴) → 𝐹 = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹𝑋)⟩}))

Proof of Theorem fnsnsplit
StepHypRef Expression
1 fnresdm 6686 . . 3 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
21adantr 480 . 2 ((𝐹 Fn 𝐴𝑋𝐴) → (𝐹𝐴) = 𝐹)
3 resundi 6010 . . 3 (𝐹 ↾ ((𝐴 ∖ {𝑋}) ∪ {𝑋})) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ (𝐹 ↾ {𝑋}))
4 difsnid 4809 . . . . 5 (𝑋𝐴 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
54adantl 481 . . . 4 ((𝐹 Fn 𝐴𝑋𝐴) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
65reseq2d 5996 . . 3 ((𝐹 Fn 𝐴𝑋𝐴) → (𝐹 ↾ ((𝐴 ∖ {𝑋}) ∪ {𝑋})) = (𝐹𝐴))
7 fnressn 7177 . . . 4 ((𝐹 Fn 𝐴𝑋𝐴) → (𝐹 ↾ {𝑋}) = {⟨𝑋, (𝐹𝑋)⟩})
87uneq2d 4167 . . 3 ((𝐹 Fn 𝐴𝑋𝐴) → ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ (𝐹 ↾ {𝑋})) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹𝑋)⟩}))
93, 6, 83eqtr3a 2800 . 2 ((𝐹 Fn 𝐴𝑋𝐴) → (𝐹𝐴) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹𝑋)⟩}))
102, 9eqtr3d 2778 1 ((𝐹 Fn 𝐴𝑋𝐴) → 𝐹 = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹𝑋)⟩}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  cdif 3947  cun 3948  {csn 4625  cop 4631  cres 5686   Fn wfn 6555  cfv 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568
This theorem is referenced by:  funresdfunsn  7210  ralxpmap  8937  reprsuc  34631  finixpnum  37613  poimirlem4  37632
  Copyright terms: Public domain W3C validator