Theorem funresdfunsndc 6402

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 5140	. . . . 5 ⊢ (Fun 𝐹 → Rel 𝐹)
2		resdmdfsn 4862	. . . . 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 1003	. . 3 ⊢ ((∀𝑥 ∈ dom 𝐹∀𝑦 ∈ dom 𝐹DECID 𝑥 = 𝑦 ∧ Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → (𝐹 ↾ (V ∖ {𝑋})) = (𝐹 ↾ (dom 𝐹 ∖ {𝑋})))
5	4	uneq1d 3229	. 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 5153	. . 3 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
7		fnsnsplitdc 6401	. . 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 1253	. 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 2175	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 819   ∧ w3a 962   = wceq 1331   ∈ wcel 1480  ∀wral 2416  Vcvv 2686   ∖ cdif 3068   ∪ cun 3069  {csn 3527  ⟨cop 3530  dom cdm 4539   ↾ cres 4541  Rel wrel 4544  Fun wfun 5117   Fn wfn 5118  ‘cfv 5123

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131

This theorem is referenced by:  strsetsid  11992