Theorem fnsnsplitdc 6558

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	⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝐹 = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩}))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝑋,𝑦

Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem fnsnsplitdc

Step	Hyp	Ref	Expression
1		fnresdm 5363	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
2	1	3ad2ant2 1021	. 2 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐹 ↾ 𝐴) = 𝐹)
3		resundi 4955	. . 3 ⊢ (𝐹 ↾ ((𝐴 ∖ {𝑋}) ∪ {𝑋})) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ (𝐹 ↾ {𝑋}))
4		dcdifsnid 6557	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑋 ∈ 𝐴) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
5	4	3adant2 1018	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
6	5	reseq2d 4942	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐹 ↾ ((𝐴 ∖ {𝑋}) ∪ {𝑋})) = (𝐹 ↾ 𝐴))
7		fnressn 5744	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐹 ↾ {𝑋}) = {⟨𝑋, (𝐹‘𝑋)⟩})
8	7	uneq2d 3313	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ (𝐹 ↾ {𝑋})) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩}))
9	8	3adant1 1017	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ (𝐹 ↾ {𝑋})) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩}))
10	3, 6, 9	3eqtr3a 2250	. 2 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐹 ↾ 𝐴) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩}))
11	2, 10	eqtr3d 2228	1 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝐹 = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩}))

Syntax hints:   → wi 4   ∧ wa 104  DECID wdc 835   ∧ w3a 980   = wceq 1364   ∈ wcel 2164  ∀wral 2472   ∖ cdif 3150   ∪ cun 3151  {csn 3618  ⟨cop 3621   ↾ cres 4661   Fn wfn 5249  ‘cfv 5254

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262

This theorem is referenced by:  funresdfunsndc  6559