Theorem funresdfunsndc 6573

Description: Restricting a function to a domain without one element of the domain of the function, and adding a pair of this element and the function value of the element results in the function itself, where equality is decidable. (Contributed by AV, 2-Dec-2018.) (Revised by Jim Kingdon, 30-Jan-2023.)

Assertion

Ref	Expression
funresdfunsndc	⊢ ((∀𝑥 ∈ dom 𝐹∀𝑦 ∈ dom 𝐹DECID 𝑥 = 𝑦 ∧ Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ((𝐹 ↾ (V ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩}) = 𝐹)

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

Proof of Theorem funresdfunsndc

Step	Hyp	Ref	Expression
1		funrel 5276	. . . . 5 ⊢ (Fun 𝐹 → Rel 𝐹)
2		resdmdfsn 4990	. . . . 5 ⊢ (Rel 𝐹 → (𝐹 ↾ (V ∖ {𝑋})) = (𝐹 ↾ (dom 𝐹 ∖ {𝑋})))
3	1, 2	syl 14	. . . 4 ⊢ (Fun 𝐹 → (𝐹 ↾ (V ∖ {𝑋})) = (𝐹 ↾ (dom 𝐹 ∖ {𝑋})))
4	3	3ad2ant2 1021	. . 3 ⊢ ((∀𝑥 ∈ dom 𝐹∀𝑦 ∈ dom 𝐹DECID 𝑥 = 𝑦 ∧ Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → (𝐹 ↾ (V ∖ {𝑋})) = (𝐹 ↾ (dom 𝐹 ∖ {𝑋})))
5	4	uneq1d 3317	. 2 ⊢ ((∀𝑥 ∈ dom 𝐹∀𝑦 ∈ dom 𝐹DECID 𝑥 = 𝑦 ∧ Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ((𝐹 ↾ (V ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩}) = ((𝐹 ↾ (dom 𝐹 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩}))
6		funfn 5289	. . 3 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
7		fnsnsplitdc 6572	. . 3 ⊢ ((∀𝑥 ∈ dom 𝐹∀𝑦 ∈ dom 𝐹DECID 𝑥 = 𝑦 ∧ 𝐹 Fn dom 𝐹 ∧ 𝑋 ∈ dom 𝐹) → 𝐹 = ((𝐹 ↾ (dom 𝐹 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩}))
8	6, 7	syl3an2b 1286	. 2 ⊢ ((∀𝑥 ∈ dom 𝐹∀𝑦 ∈ dom 𝐹DECID 𝑥 = 𝑦 ∧ Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → 𝐹 = ((𝐹 ↾ (dom 𝐹 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩}))
9	5, 8	eqtr4d 2232	1 ⊢ ((∀𝑥 ∈ dom 𝐹∀𝑦 ∈ dom 𝐹DECID 𝑥 = 𝑦 ∧ Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ((𝐹 ↾ (V ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩}) = 𝐹)

Syntax hints:   → wi 4  DECID wdc 835   ∧ w3a 980   = wceq 1364   ∈ wcel 2167  ∀wral 2475  Vcvv 2763   ∖ cdif 3154   ∪ cun 3155  {csn 3623  ⟨cop 3626  dom cdm 4664   ↾ cres 4666  Rel wrel 4669  Fun wfun 5253   Fn wfn 5254  ‘cfv 5259

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267

This theorem is referenced by:  strsetsid  12736