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Theorem fnsnsplitdc 6505
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 5325 . . 3 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
213ad2ant2 1019 . 2 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐹 Fn 𝐴𝑋𝐴) → (𝐹𝐴) = 𝐹)
3 resundi 4920 . . 3 (𝐹 ↾ ((𝐴 ∖ {𝑋}) ∪ {𝑋})) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ (𝐹 ↾ {𝑋}))
4 dcdifsnid 6504 . . . . 5 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝑋𝐴) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
543adant2 1016 . . . 4 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐹 Fn 𝐴𝑋𝐴) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
65reseq2d 4907 . . 3 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐹 Fn 𝐴𝑋𝐴) → (𝐹 ↾ ((𝐴 ∖ {𝑋}) ∪ {𝑋})) = (𝐹𝐴))
7 fnressn 5702 . . . . 5 ((𝐹 Fn 𝐴𝑋𝐴) → (𝐹 ↾ {𝑋}) = {⟨𝑋, (𝐹𝑋)⟩})
87uneq2d 3289 . . . 4 ((𝐹 Fn 𝐴𝑋𝐴) → ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ (𝐹 ↾ {𝑋})) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹𝑋)⟩}))
983adant1 1015 . . 3 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐹 Fn 𝐴𝑋𝐴) → ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ (𝐹 ↾ {𝑋})) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹𝑋)⟩}))
103, 6, 93eqtr3a 2234 . 2 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐹 Fn 𝐴𝑋𝐴) → (𝐹𝐴) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹𝑋)⟩}))
112, 10eqtr3d 2212 1 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐹 Fn 𝐴𝑋𝐴) → 𝐹 = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹𝑋)⟩}))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  DECID wdc 834  w3a 978   = wceq 1353  wcel 2148  wral 2455  cdif 3126  cun 3127  {csn 3592  cop 3595  cres 4628   Fn wfn 5211  cfv 5216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224
This theorem is referenced by:  funresdfunsndc  6506
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