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Theorem fnsnsplit 6615
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 6161 . . 3 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
21adantr 472 . 2 ((𝐹 Fn 𝐴𝑋𝐴) → (𝐹𝐴) = 𝐹)
3 resundi 5568 . . 3 (𝐹 ↾ ((𝐴 ∖ {𝑋}) ∪ {𝑋})) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ (𝐹 ↾ {𝑋}))
4 difsnid 4486 . . . . 5 (𝑋𝐴 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
54adantl 473 . . . 4 ((𝐹 Fn 𝐴𝑋𝐴) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
65reseq2d 5551 . . 3 ((𝐹 Fn 𝐴𝑋𝐴) → (𝐹 ↾ ((𝐴 ∖ {𝑋}) ∪ {𝑋})) = (𝐹𝐴))
7 fnressn 6589 . . . 4 ((𝐹 Fn 𝐴𝑋𝐴) → (𝐹 ↾ {𝑋}) = {⟨𝑋, (𝐹𝑋)⟩})
87uneq2d 3910 . . 3 ((𝐹 Fn 𝐴𝑋𝐴) → ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ (𝐹 ↾ {𝑋})) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹𝑋)⟩}))
93, 6, 83eqtr3a 2818 . 2 ((𝐹 Fn 𝐴𝑋𝐴) → (𝐹𝐴) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹𝑋)⟩}))
102, 9eqtr3d 2796 1 ((𝐹 Fn 𝐴𝑋𝐴) → 𝐹 = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹𝑋)⟩}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  cdif 3712  cun 3713  {csn 4321  cop 4327  cres 5268   Fn wfn 6044  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057
This theorem is referenced by:  funresdfunsn  6620  ralxpmap  8075  reprsuc  31023  finixpnum  33725  poimirlem4  33744
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