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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 𝐹𝑦 ∈ dom 𝐹DECID 𝑥 = 𝑦 ∧ Fun 𝐹𝑋 ∈ dom 𝐹) → ((𝐹 ↾ (V ∖ {𝑋})) ∪ {⟨𝑋, (𝐹𝑋)⟩}) = 𝐹)
Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝑋,𝑦

Proof of Theorem funresdfunsndc
StepHypRef Expression
1 funrel 5140 . . . . 5 (Fun 𝐹 → Rel 𝐹)
2 resdmdfsn 4862 . . . . 5 (Rel 𝐹 → (𝐹 ↾ (V ∖ {𝑋})) = (𝐹 ↾ (dom 𝐹 ∖ {𝑋})))
31, 2syl 14 . . . 4 (Fun 𝐹 → (𝐹 ↾ (V ∖ {𝑋})) = (𝐹 ↾ (dom 𝐹 ∖ {𝑋})))
433ad2ant2 1003 . . 3 ((∀𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹DECID 𝑥 = 𝑦 ∧ Fun 𝐹𝑋 ∈ dom 𝐹) → (𝐹 ↾ (V ∖ {𝑋})) = (𝐹 ↾ (dom 𝐹 ∖ {𝑋})))
54uneq1d 3229 . 2 ((∀𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹DECID 𝑥 = 𝑦 ∧ Fun 𝐹𝑋 ∈ dom 𝐹) → ((𝐹 ↾ (V ∖ {𝑋})) ∪ {⟨𝑋, (𝐹𝑋)⟩}) = ((𝐹 ↾ (dom 𝐹 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹𝑋)⟩}))
6 funfn 5153 . . 3 (Fun 𝐹𝐹 Fn dom 𝐹)
7 fnsnsplitdc 6401 . . 3 ((∀𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹DECID 𝑥 = 𝑦𝐹 Fn dom 𝐹𝑋 ∈ dom 𝐹) → 𝐹 = ((𝐹 ↾ (dom 𝐹 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹𝑋)⟩}))
86, 7syl3an2b 1253 . 2 ((∀𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹DECID 𝑥 = 𝑦 ∧ Fun 𝐹𝑋 ∈ dom 𝐹) → 𝐹 = ((𝐹 ↾ (dom 𝐹 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹𝑋)⟩}))
95, 8eqtr4d 2175 1 ((∀𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹DECID 𝑥 = 𝑦 ∧ Fun 𝐹𝑋 ∈ dom 𝐹) → ((𝐹 ↾ (V ∖ {𝑋})) ∪ {⟨𝑋, (𝐹𝑋)⟩}) = 𝐹)
Colors of variables: wff set class
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  12006
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