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Theorem fsnunfv 5587
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 4808 . . . . . . . . 9 dom (𝐹 ↾ {𝑋}) = ({𝑋} ∩ dom 𝐹)
2 incom 3236 . . . . . . . . 9 ({𝑋} ∩ dom 𝐹) = (dom 𝐹 ∩ {𝑋})
31, 2eqtri 2136 . . . . . . . 8 dom (𝐹 ↾ {𝑋}) = (dom 𝐹 ∩ {𝑋})
4 disjsn 3553 . . . . . . . . 9 ((dom 𝐹 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ dom 𝐹)
54biimpri 132 . . . . . . . 8 𝑋 ∈ dom 𝐹 → (dom 𝐹 ∩ {𝑋}) = ∅)
63, 5syl5eq 2160 . . . . . . 7 𝑋 ∈ dom 𝐹 → dom (𝐹 ↾ {𝑋}) = ∅)
763ad2ant3 987 . . . . . 6 ((𝑋𝑉𝑌𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → dom (𝐹 ↾ {𝑋}) = ∅)
8 relres 4815 . . . . . . 7 Rel (𝐹 ↾ {𝑋})
9 reldm0 4725 . . . . . . 7 (Rel (𝐹 ↾ {𝑋}) → ((𝐹 ↾ {𝑋}) = ∅ ↔ dom (𝐹 ↾ {𝑋}) = ∅))
108, 9ax-mp 5 . . . . . 6 ((𝐹 ↾ {𝑋}) = ∅ ↔ dom (𝐹 ↾ {𝑋}) = ∅)
117, 10sylibr 133 . . . . 5 ((𝑋𝑉𝑌𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → (𝐹 ↾ {𝑋}) = ∅)
12 fnsng 5138 . . . . . . 7 ((𝑋𝑉𝑌𝑊) → {⟨𝑋, 𝑌⟩} Fn {𝑋})
13123adant3 984 . . . . . 6 ((𝑋𝑉𝑌𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → {⟨𝑋, 𝑌⟩} Fn {𝑋})
14 fnresdm 5200 . . . . . 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 3199 . . . 4 ((𝑋𝑉𝑌𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → ((𝐹 ↾ {𝑋}) ∪ ({⟨𝑋, 𝑌⟩} ↾ {𝑋})) = (∅ ∪ {⟨𝑋, 𝑌⟩}))
17 resundir 4801 . . . 4 ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ {𝑋}) = ((𝐹 ↾ {𝑋}) ∪ ({⟨𝑋, 𝑌⟩} ↾ {𝑋}))
18 uncom 3188 . . . . 5 (∅ ∪ {⟨𝑋, 𝑌⟩}) = ({⟨𝑋, 𝑌⟩} ∪ ∅)
19 un0 3364 . . . . 5 ({⟨𝑋, 𝑌⟩} ∪ ∅) = {⟨𝑋, 𝑌⟩}
2018, 19eqtr2i 2137 . . . 4 {⟨𝑋, 𝑌⟩} = (∅ ∪ {⟨𝑋, 𝑌⟩})
2116, 17, 203eqtr4g 2173 . . 3 ((𝑋𝑉𝑌𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ {𝑋}) = {⟨𝑋, 𝑌⟩})
2221fveq1d 5389 . 2 ((𝑋𝑉𝑌𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → (((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ {𝑋})‘𝑋) = ({⟨𝑋, 𝑌⟩}‘𝑋))
23 snidg 3522 . . . 4 (𝑋𝑉𝑋 ∈ {𝑋})
24233ad2ant1 985 . . 3 ((𝑋𝑉𝑌𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → 𝑋 ∈ {𝑋})
25 fvres 5411 . . 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 5582 . . 3 ((𝑋𝑉𝑌𝑊) → ({⟨𝑋, 𝑌⟩}‘𝑋) = 𝑌)
28273adant3 984 . 2 ((𝑋𝑉𝑌𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → ({⟨𝑋, 𝑌⟩}‘𝑋) = 𝑌)
2922, 26, 283eqtr3d 2156 1 ((𝑋𝑉𝑌𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩})‘𝑋) = 𝑌)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 104  w3a 945   = wceq 1314  wcel 1463  cun 3037  cin 3038  c0 3331  {csn 3495  cop 3498  dom cdm 4507  cres 4509  Rel wrel 4512   Fn wfn 5086  cfv 5091
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-res 4519  df-iota 5056  df-fun 5093  df-fn 5094  df-fv 5099
This theorem is referenced by:  tfrlemisucaccv  6188  tfr1onlemsucaccv  6204  tfrcllemsucaccv  6217  inftonninf  10154  hashinfom  10464  zfz1isolemiso  10522  fvsetsid  11888
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