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Theorem fsnunfv 5718
Description: Recover the added point from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by NM, 18-May-2017.)
Assertion
Ref Expression
fsnunfv ((𝑋𝑉𝑌𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩})‘𝑋) = 𝑌)

Proof of Theorem fsnunfv
StepHypRef Expression
1 dmres 4929 . . . . . . . . 9 dom (𝐹 ↾ {𝑋}) = ({𝑋} ∩ dom 𝐹)
2 incom 3328 . . . . . . . . 9 ({𝑋} ∩ dom 𝐹) = (dom 𝐹 ∩ {𝑋})
31, 2eqtri 2198 . . . . . . . 8 dom (𝐹 ↾ {𝑋}) = (dom 𝐹 ∩ {𝑋})
4 disjsn 3655 . . . . . . . . 9 ((dom 𝐹 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ dom 𝐹)
54biimpri 133 . . . . . . . 8 𝑋 ∈ dom 𝐹 → (dom 𝐹 ∩ {𝑋}) = ∅)
63, 5eqtrid 2222 . . . . . . 7 𝑋 ∈ dom 𝐹 → dom (𝐹 ↾ {𝑋}) = ∅)
763ad2ant3 1020 . . . . . 6 ((𝑋𝑉𝑌𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → dom (𝐹 ↾ {𝑋}) = ∅)
8 relres 4936 . . . . . . 7 Rel (𝐹 ↾ {𝑋})
9 reldm0 4846 . . . . . . 7 (Rel (𝐹 ↾ {𝑋}) → ((𝐹 ↾ {𝑋}) = ∅ ↔ dom (𝐹 ↾ {𝑋}) = ∅))
108, 9ax-mp 5 . . . . . 6 ((𝐹 ↾ {𝑋}) = ∅ ↔ dom (𝐹 ↾ {𝑋}) = ∅)
117, 10sylibr 134 . . . . 5 ((𝑋𝑉𝑌𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → (𝐹 ↾ {𝑋}) = ∅)
12 fnsng 5264 . . . . . . 7 ((𝑋𝑉𝑌𝑊) → {⟨𝑋, 𝑌⟩} Fn {𝑋})
13123adant3 1017 . . . . . 6 ((𝑋𝑉𝑌𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → {⟨𝑋, 𝑌⟩} Fn {𝑋})
14 fnresdm 5326 . . . . . 6 ({⟨𝑋, 𝑌⟩} Fn {𝑋} → ({⟨𝑋, 𝑌⟩} ↾ {𝑋}) = {⟨𝑋, 𝑌⟩})
1513, 14syl 14 . . . . 5 ((𝑋𝑉𝑌𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → ({⟨𝑋, 𝑌⟩} ↾ {𝑋}) = {⟨𝑋, 𝑌⟩})
1611, 15uneq12d 3291 . . . 4 ((𝑋𝑉𝑌𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → ((𝐹 ↾ {𝑋}) ∪ ({⟨𝑋, 𝑌⟩} ↾ {𝑋})) = (∅ ∪ {⟨𝑋, 𝑌⟩}))
17 resundir 4922 . . . 4 ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ {𝑋}) = ((𝐹 ↾ {𝑋}) ∪ ({⟨𝑋, 𝑌⟩} ↾ {𝑋}))
18 uncom 3280 . . . . 5 (∅ ∪ {⟨𝑋, 𝑌⟩}) = ({⟨𝑋, 𝑌⟩} ∪ ∅)
19 un0 3457 . . . . 5 ({⟨𝑋, 𝑌⟩} ∪ ∅) = {⟨𝑋, 𝑌⟩}
2018, 19eqtr2i 2199 . . . 4 {⟨𝑋, 𝑌⟩} = (∅ ∪ {⟨𝑋, 𝑌⟩})
2116, 17, 203eqtr4g 2235 . . 3 ((𝑋𝑉𝑌𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ {𝑋}) = {⟨𝑋, 𝑌⟩})
2221fveq1d 5518 . 2 ((𝑋𝑉𝑌𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → (((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ {𝑋})‘𝑋) = ({⟨𝑋, 𝑌⟩}‘𝑋))
23 snidg 3622 . . . 4 (𝑋𝑉𝑋 ∈ {𝑋})
24233ad2ant1 1018 . . 3 ((𝑋𝑉𝑌𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → 𝑋 ∈ {𝑋})
25 fvres 5540 . . 3 (𝑋 ∈ {𝑋} → (((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ {𝑋})‘𝑋) = ((𝐹 ∪ {⟨𝑋, 𝑌⟩})‘𝑋))
2624, 25syl 14 . 2 ((𝑋𝑉𝑌𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → (((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ {𝑋})‘𝑋) = ((𝐹 ∪ {⟨𝑋, 𝑌⟩})‘𝑋))
27 fvsng 5713 . . 3 ((𝑋𝑉𝑌𝑊) → ({⟨𝑋, 𝑌⟩}‘𝑋) = 𝑌)
28273adant3 1017 . 2 ((𝑋𝑉𝑌𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → ({⟨𝑋, 𝑌⟩}‘𝑋) = 𝑌)
2922, 26, 283eqtr3d 2218 1 ((𝑋𝑉𝑌𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩})‘𝑋) = 𝑌)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 105  w3a 978   = wceq 1353  wcel 2148  cun 3128  cin 3129  c0 3423  {csn 3593  cop 3596  dom cdm 4627  cres 4629  Rel wrel 4632   Fn wfn 5212  cfv 5217
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 4122  ax-pow 4175  ax-pr 4210
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  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-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-res 4639  df-iota 5179  df-fun 5219  df-fn 5220  df-fv 5225
This theorem is referenced by:  tfrlemisucaccv  6326  tfr1onlemsucaccv  6342  tfrcllemsucaccv  6355  inftonninf  10441  hashinfom  10758  zfz1isolemiso  10819  fvsetsid  12496
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