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Theorem fnsnsplitdc 6410
 Description: Split a function into a single point and all the rest. (Contributed by Stefan O'Rear, 27-Feb-2015.) (Revised by Jim Kingdon, 29-Jan-2023.)
Assertion
Ref Expression
fnsnsplitdc ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐹 Fn 𝐴𝑋𝐴) → 𝐹 = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹𝑋)⟩}))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝑋,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem fnsnsplitdc
StepHypRef Expression
1 fnresdm 5241 . . 3 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
213ad2ant2 1004 . 2 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐹 Fn 𝐴𝑋𝐴) → (𝐹𝐴) = 𝐹)
3 resundi 4841 . . 3 (𝐹 ↾ ((𝐴 ∖ {𝑋}) ∪ {𝑋})) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ (𝐹 ↾ {𝑋}))
4 dcdifsnid 6409 . . . . 5 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝑋𝐴) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
543adant2 1001 . . . 4 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐹 Fn 𝐴𝑋𝐴) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
65reseq2d 4828 . . 3 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐹 Fn 𝐴𝑋𝐴) → (𝐹 ↾ ((𝐴 ∖ {𝑋}) ∪ {𝑋})) = (𝐹𝐴))
7 fnressn 5615 . . . . 5 ((𝐹 Fn 𝐴𝑋𝐴) → (𝐹 ↾ {𝑋}) = {⟨𝑋, (𝐹𝑋)⟩})
87uneq2d 3236 . . . 4 ((𝐹 Fn 𝐴𝑋𝐴) → ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ (𝐹 ↾ {𝑋})) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹𝑋)⟩}))
983adant1 1000 . . 3 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐹 Fn 𝐴𝑋𝐴) → ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ (𝐹 ↾ {𝑋})) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹𝑋)⟩}))
103, 6, 93eqtr3a 2197 . 2 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐹 Fn 𝐴𝑋𝐴) → (𝐹𝐴) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹𝑋)⟩}))
112, 10eqtr3d 2175 1 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐹 Fn 𝐴𝑋𝐴) → 𝐹 = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹𝑋)⟩}))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103  DECID wdc 820   ∧ w3a 963   = wceq 1332   ∈ wcel 1481  ∀wral 2417   ∖ cdif 3074   ∪ cun 3075  {csn 3533  ⟨cop 3536   ↾ cres 4550   Fn wfn 5127  ‘cfv 5132 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4055  ax-pow 4107  ax-pr 4140 This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-reu 2424  df-v 2692  df-sbc 2915  df-dif 3079  df-un 3081  df-in 3083  df-ss 3090  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-br 3939  df-opab 3999  df-id 4224  df-xp 4554  df-rel 4555  df-cnv 4556  df-co 4557  df-dm 4558  df-rn 4559  df-res 4560  df-ima 4561  df-iota 5097  df-fun 5134  df-fn 5135  df-f 5136  df-f1 5137  df-fo 5138  df-f1o 5139  df-fv 5140 This theorem is referenced by:  funresdfunsndc  6411
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