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Theorem fsnunfv 5288
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 ((𝑋 𝑉 𝑌 𝑊 ¬ 𝑋 dom 𝐹) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩})‘𝑋) = 𝑌)

Proof of Theorem fsnunfv
StepHypRef Expression
1 dmres 4559 . . . . . . . . 9 dom (𝐹 ↾ {𝑋}) = ({𝑋} ∩ dom 𝐹)
2 incom 3106 . . . . . . . . 9 ({𝑋} ∩ dom 𝐹) = (dom 𝐹 ∩ {𝑋})
31, 2eqtri 2042 . . . . . . . 8 dom (𝐹 ↾ {𝑋}) = (dom 𝐹 ∩ {𝑋})
4 disjsn 3406 . . . . . . . . 9 ((dom 𝐹 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 dom 𝐹)
54biimpri 124 . . . . . . . 8 𝑋 dom 𝐹 → (dom 𝐹 ∩ {𝑋}) = ∅)
63, 5syl5eq 2066 . . . . . . 7 𝑋 dom 𝐹 → dom (𝐹 ↾ {𝑋}) = ∅)
763ad2ant3 915 . . . . . 6 ((𝑋 𝑉 𝑌 𝑊 ¬ 𝑋 dom 𝐹) → dom (𝐹 ↾ {𝑋}) = ∅)
8 relres 4566 . . . . . . 7 Rel (𝐹 ↾ {𝑋})
9 reldm0 4480 . . . . . . 7 (Rel (𝐹 ↾ {𝑋}) → ((𝐹 ↾ {𝑋}) = ∅ ↔ dom (𝐹 ↾ {𝑋}) = ∅))
108, 9ax-mp 7 . . . . . 6 ((𝐹 ↾ {𝑋}) = ∅ ↔ dom (𝐹 ↾ {𝑋}) = ∅)
117, 10sylibr 137 . . . . 5 ((𝑋 𝑉 𝑌 𝑊 ¬ 𝑋 dom 𝐹) → (𝐹 ↾ {𝑋}) = ∅)
12 fnsng 4873 . . . . . . 7 ((𝑋 𝑉 𝑌 𝑊) → {⟨𝑋, 𝑌⟩} Fn {𝑋})
13123adant3 912 . . . . . 6 ((𝑋 𝑉 𝑌 𝑊 ¬ 𝑋 dom 𝐹) → {⟨𝑋, 𝑌⟩} Fn {𝑋})
14 fnresdm 4934 . . . . . 6 ({⟨𝑋, 𝑌⟩} Fn {𝑋} → ({⟨𝑋, 𝑌⟩} ↾ {𝑋}) = {⟨𝑋, 𝑌⟩})
1513, 14syl 14 . . . . 5 ((𝑋 𝑉 𝑌 𝑊 ¬ 𝑋 dom 𝐹) → ({⟨𝑋, 𝑌⟩} ↾ {𝑋}) = {⟨𝑋, 𝑌⟩})
1611, 15uneq12d 3075 . . . 4 ((𝑋 𝑉 𝑌 𝑊 ¬ 𝑋 dom 𝐹) → ((𝐹 ↾ {𝑋}) ∪ ({⟨𝑋, 𝑌⟩} ↾ {𝑋})) = (∅ ∪ {⟨𝑋, 𝑌⟩}))
17 resundir 4553 . . . 4 ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ {𝑋}) = ((𝐹 ↾ {𝑋}) ∪ ({⟨𝑋, 𝑌⟩} ↾ {𝑋}))
18 uncom 3064 . . . . 5 (∅ ∪ {⟨𝑋, 𝑌⟩}) = ({⟨𝑋, 𝑌⟩} ∪ ∅)
19 un0 3228 . . . . 5 ({⟨𝑋, 𝑌⟩} ∪ ∅) = {⟨𝑋, 𝑌⟩}
2018, 19eqtr2i 2043 . . . 4 {⟨𝑋, 𝑌⟩} = (∅ ∪ {⟨𝑋, 𝑌⟩})
2116, 17, 203eqtr4g 2079 . . 3 ((𝑋 𝑉 𝑌 𝑊 ¬ 𝑋 dom 𝐹) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ {𝑋}) = {⟨𝑋, 𝑌⟩})
2221fveq1d 5105 . 2 ((𝑋 𝑉 𝑌 𝑊 ¬ 𝑋 dom 𝐹) → (((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ {𝑋})‘𝑋) = ({⟨𝑋, 𝑌⟩}‘𝑋))
23 snidg 3375 . . . 4 (𝑋 𝑉𝑋 {𝑋})
24233ad2ant1 913 . . 3 ((𝑋 𝑉 𝑌 𝑊 ¬ 𝑋 dom 𝐹) → 𝑋 {𝑋})
25 fvres 5123 . . 3 (𝑋 {𝑋} → (((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ {𝑋})‘𝑋) = ((𝐹 ∪ {⟨𝑋, 𝑌⟩})‘𝑋))
2624, 25syl 14 . 2 ((𝑋 𝑉 𝑌 𝑊 ¬ 𝑋 dom 𝐹) → (((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ {𝑋})‘𝑋) = ((𝐹 ∪ {⟨𝑋, 𝑌⟩})‘𝑋))
27 fvsng 5284 . . 3 ((𝑋 𝑉 𝑌 𝑊) → ({⟨𝑋, 𝑌⟩}‘𝑋) = 𝑌)
28273adant3 912 . 2 ((𝑋 𝑉 𝑌 𝑊 ¬ 𝑋 dom 𝐹) → ({⟨𝑋, 𝑌⟩}‘𝑋) = 𝑌)
2922, 26, 283eqtr3d 2062 1 ((𝑋 𝑉 𝑌 𝑊 ¬ 𝑋 dom 𝐹) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩})‘𝑋) = 𝑌)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 98   w3a 873   = wceq 1228   wcel 1374  cun 2892  cin 2893  c0 3201  {csn 3350  cop 3353  dom cdm 4272  cres 4274  Rel wrel 4277   Fn wfn 4824  cfv 4829
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-res 4284  df-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837
This theorem is referenced by:  tfrlemisucaccv  5860
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