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Theorem fsnunfv 5306
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 4575 . . . . . . . . 9 dom (𝐹 ↾ {𝑋}) = ({𝑋} ∩ dom 𝐹)
2 incom 3123 . . . . . . . . 9 ({𝑋} ∩ dom 𝐹) = (dom 𝐹 ∩ {𝑋})
31, 2eqtri 2057 . . . . . . . 8 dom (𝐹 ↾ {𝑋}) = (dom 𝐹 ∩ {𝑋})
4 disjsn 3423 . . . . . . . . 9 ((dom 𝐹 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 dom 𝐹)
54biimpri 124 . . . . . . . 8 𝑋 dom 𝐹 → (dom 𝐹 ∩ {𝑋}) = ∅)
63, 5syl5eq 2081 . . . . . . 7 𝑋 dom 𝐹 → dom (𝐹 ↾ {𝑋}) = ∅)
763ad2ant3 926 . . . . . 6 ((𝑋 𝑉 𝑌 𝑊 ¬ 𝑋 dom 𝐹) → dom (𝐹 ↾ {𝑋}) = ∅)
8 relres 4582 . . . . . . 7 Rel (𝐹 ↾ {𝑋})
9 reldm0 4496 . . . . . . 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 4890 . . . . . . 7 ((𝑋 𝑉 𝑌 𝑊) → {⟨𝑋, 𝑌⟩} Fn {𝑋})
13123adant3 923 . . . . . 6 ((𝑋 𝑉 𝑌 𝑊 ¬ 𝑋 dom 𝐹) → {⟨𝑋, 𝑌⟩} Fn {𝑋})
14 fnresdm 4951 . . . . . 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 3092 . . . 4 ((𝑋 𝑉 𝑌 𝑊 ¬ 𝑋 dom 𝐹) → ((𝐹 ↾ {𝑋}) ∪ ({⟨𝑋, 𝑌⟩} ↾ {𝑋})) = (∅ ∪ {⟨𝑋, 𝑌⟩}))
17 resundir 4569 . . . 4 ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ {𝑋}) = ((𝐹 ↾ {𝑋}) ∪ ({⟨𝑋, 𝑌⟩} ↾ {𝑋}))
18 uncom 3081 . . . . 5 (∅ ∪ {⟨𝑋, 𝑌⟩}) = ({⟨𝑋, 𝑌⟩} ∪ ∅)
19 un0 3245 . . . . 5 ({⟨𝑋, 𝑌⟩} ∪ ∅) = {⟨𝑋, 𝑌⟩}
2018, 19eqtr2i 2058 . . . 4 {⟨𝑋, 𝑌⟩} = (∅ ∪ {⟨𝑋, 𝑌⟩})
2116, 17, 203eqtr4g 2094 . . 3 ((𝑋 𝑉 𝑌 𝑊 ¬ 𝑋 dom 𝐹) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ {𝑋}) = {⟨𝑋, 𝑌⟩})
2221fveq1d 5123 . 2 ((𝑋 𝑉 𝑌 𝑊 ¬ 𝑋 dom 𝐹) → (((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ {𝑋})‘𝑋) = ({⟨𝑋, 𝑌⟩}‘𝑋))
23 snidg 3392 . . . 4 (𝑋 𝑉𝑋 {𝑋})
24233ad2ant1 924 . . 3 ((𝑋 𝑉 𝑌 𝑊 ¬ 𝑋 dom 𝐹) → 𝑋 {𝑋})
25 fvres 5141 . . 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 5302 . . 3 ((𝑋 𝑉 𝑌 𝑊) → ({⟨𝑋, 𝑌⟩}‘𝑋) = 𝑌)
28273adant3 923 . 2 ((𝑋 𝑉 𝑌 𝑊 ¬ 𝑋 dom 𝐹) → ({⟨𝑋, 𝑌⟩}‘𝑋) = 𝑌)
2922, 26, 283eqtr3d 2077 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 884   = wceq 1242   wcel 1390  cun 2909  cin 2910  c0 3218  {csn 3367  cop 3370  dom cdm 4288  cres 4290  Rel wrel 4293   Fn wfn 4840  cfv 4845
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-res 4300  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853
This theorem is referenced by:  tfrlemisucaccv  5880
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