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Theorem fsnunfv 6674
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 5622 . . . . . . . . 9 dom (𝐹 ↾ {𝑋}) = ({𝑋} ∩ dom 𝐹)
2 incom 4004 . . . . . . . . 9 ({𝑋} ∩ dom 𝐹) = (dom 𝐹 ∩ {𝑋})
31, 2eqtri 2828 . . . . . . . 8 dom (𝐹 ↾ {𝑋}) = (dom 𝐹 ∩ {𝑋})
4 disjsn 4438 . . . . . . . . 9 ((dom 𝐹 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ dom 𝐹)
54biimpri 219 . . . . . . . 8 𝑋 ∈ dom 𝐹 → (dom 𝐹 ∩ {𝑋}) = ∅)
63, 5syl5eq 2852 . . . . . . 7 𝑋 ∈ dom 𝐹 → dom (𝐹 ↾ {𝑋}) = ∅)
763ad2ant3 1158 . . . . . 6 ((𝑋𝑉𝑌𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → dom (𝐹 ↾ {𝑋}) = ∅)
8 relres 5629 . . . . . . 7 Rel (𝐹 ↾ {𝑋})
9 reldm0 5544 . . . . . . 7 (Rel (𝐹 ↾ {𝑋}) → ((𝐹 ↾ {𝑋}) = ∅ ↔ dom (𝐹 ↾ {𝑋}) = ∅))
108, 9ax-mp 5 . . . . . 6 ((𝐹 ↾ {𝑋}) = ∅ ↔ dom (𝐹 ↾ {𝑋}) = ∅)
117, 10sylibr 225 . . . . 5 ((𝑋𝑉𝑌𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → (𝐹 ↾ {𝑋}) = ∅)
12 fnsng 6148 . . . . . . 7 ((𝑋𝑉𝑌𝑊) → {⟨𝑋, 𝑌⟩} Fn {𝑋})
13123adant3 1155 . . . . . 6 ((𝑋𝑉𝑌𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → {⟨𝑋, 𝑌⟩} Fn {𝑋})
14 fnresdm 6207 . . . . . 6 ({⟨𝑋, 𝑌⟩} Fn {𝑋} → ({⟨𝑋, 𝑌⟩} ↾ {𝑋}) = {⟨𝑋, 𝑌⟩})
1513, 14syl 17 . . . . 5 ((𝑋𝑉𝑌𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → ({⟨𝑋, 𝑌⟩} ↾ {𝑋}) = {⟨𝑋, 𝑌⟩})
1611, 15uneq12d 3967 . . . 4 ((𝑋𝑉𝑌𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → ((𝐹 ↾ {𝑋}) ∪ ({⟨𝑋, 𝑌⟩} ↾ {𝑋})) = (∅ ∪ {⟨𝑋, 𝑌⟩}))
17 resundir 5615 . . . 4 ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ {𝑋}) = ((𝐹 ↾ {𝑋}) ∪ ({⟨𝑋, 𝑌⟩} ↾ {𝑋}))
18 uncom 3956 . . . . 5 (∅ ∪ {⟨𝑋, 𝑌⟩}) = ({⟨𝑋, 𝑌⟩} ∪ ∅)
19 un0 4165 . . . . 5 ({⟨𝑋, 𝑌⟩} ∪ ∅) = {⟨𝑋, 𝑌⟩}
2018, 19eqtr2i 2829 . . . 4 {⟨𝑋, 𝑌⟩} = (∅ ∪ {⟨𝑋, 𝑌⟩})
2116, 17, 203eqtr4g 2865 . . 3 ((𝑋𝑉𝑌𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ {𝑋}) = {⟨𝑋, 𝑌⟩})
2221fveq1d 6406 . 2 ((𝑋𝑉𝑌𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → (((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ {𝑋})‘𝑋) = ({⟨𝑋, 𝑌⟩}‘𝑋))
23 snidg 4400 . . . 4 (𝑋𝑉𝑋 ∈ {𝑋})
24233ad2ant1 1156 . . 3 ((𝑋𝑉𝑌𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → 𝑋 ∈ {𝑋})
25 fvres 6423 . . 3 (𝑋 ∈ {𝑋} → (((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ {𝑋})‘𝑋) = ((𝐹 ∪ {⟨𝑋, 𝑌⟩})‘𝑋))
2624, 25syl 17 . 2 ((𝑋𝑉𝑌𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → (((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ {𝑋})‘𝑋) = ((𝐹 ∪ {⟨𝑋, 𝑌⟩})‘𝑋))
27 fvsng 6668 . . 3 ((𝑋𝑉𝑌𝑊) → ({⟨𝑋, 𝑌⟩}‘𝑋) = 𝑌)
28273adant3 1155 . 2 ((𝑋𝑉𝑌𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → ({⟨𝑋, 𝑌⟩}‘𝑋) = 𝑌)
2922, 26, 283eqtr3d 2848 1 ((𝑋𝑉𝑌𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩})‘𝑋) = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  w3a 1100   = wceq 1637  wcel 2156  cun 3767  cin 3768  c0 4116  {csn 4370  cop 4376  dom cdm 5311  cres 5313  Rel wrel 5316   Fn wfn 6092  cfv 6097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-res 5323  df-iota 6060  df-fun 6099  df-fn 6100  df-fv 6105
This theorem is referenced by:  hashf1lem1  13452  cats1un  13695  fvsetsid  16097  islindf4  20383  wlkp1lem3  26796  wlkp1lem7  26800  wlkp1lem8  26801  eupth2eucrct  27386  mapfzcons2  37778  fnchoice  39676  nnsum4primeseven  42257  nnsum4primesevenALTV  42258
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