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Theorem fvunsng 5300
Description: Remove an ordered pair not participating in a function value. (Contributed by Jim Kingdon, 7-Jan-2019.)
Assertion
Ref Expression
fvunsng ((𝐷 𝑉 B𝐷) → ((A ∪ {⟨B, 𝐶⟩})‘𝐷) = (A𝐷))

Proof of Theorem fvunsng
StepHypRef Expression
1 snidg 3392 . . . 4 (𝐷 𝑉𝐷 {𝐷})
2 fvres 5141 . . . 4 (𝐷 {𝐷} → (((A ∪ {⟨B, 𝐶⟩}) ↾ {𝐷})‘𝐷) = ((A ∪ {⟨B, 𝐶⟩})‘𝐷))
31, 2syl 14 . . 3 (𝐷 𝑉 → (((A ∪ {⟨B, 𝐶⟩}) ↾ {𝐷})‘𝐷) = ((A ∪ {⟨B, 𝐶⟩})‘𝐷))
4 resundir 4569 . . . . 5 ((A ∪ {⟨B, 𝐶⟩}) ↾ {𝐷}) = ((A ↾ {𝐷}) ∪ ({⟨B, 𝐶⟩} ↾ {𝐷}))
5 elsni 3391 . . . . . . . . 9 (B {𝐷} → B = 𝐷)
65necon3ai 2248 . . . . . . . 8 (B𝐷 → ¬ B {𝐷})
7 ressnop0 5287 . . . . . . . 8 B {𝐷} → ({⟨B, 𝐶⟩} ↾ {𝐷}) = ∅)
86, 7syl 14 . . . . . . 7 (B𝐷 → ({⟨B, 𝐶⟩} ↾ {𝐷}) = ∅)
98uneq2d 3091 . . . . . 6 (B𝐷 → ((A ↾ {𝐷}) ∪ ({⟨B, 𝐶⟩} ↾ {𝐷})) = ((A ↾ {𝐷}) ∪ ∅))
10 un0 3245 . . . . . 6 ((A ↾ {𝐷}) ∪ ∅) = (A ↾ {𝐷})
119, 10syl6eq 2085 . . . . 5 (B𝐷 → ((A ↾ {𝐷}) ∪ ({⟨B, 𝐶⟩} ↾ {𝐷})) = (A ↾ {𝐷}))
124, 11syl5eq 2081 . . . 4 (B𝐷 → ((A ∪ {⟨B, 𝐶⟩}) ↾ {𝐷}) = (A ↾ {𝐷}))
1312fveq1d 5123 . . 3 (B𝐷 → (((A ∪ {⟨B, 𝐶⟩}) ↾ {𝐷})‘𝐷) = ((A ↾ {𝐷})‘𝐷))
143, 13sylan9req 2090 . 2 ((𝐷 𝑉 B𝐷) → ((A ∪ {⟨B, 𝐶⟩})‘𝐷) = ((A ↾ {𝐷})‘𝐷))
15 fvres 5141 . . . 4 (𝐷 {𝐷} → ((A ↾ {𝐷})‘𝐷) = (A𝐷))
161, 15syl 14 . . 3 (𝐷 𝑉 → ((A ↾ {𝐷})‘𝐷) = (A𝐷))
1716adantr 261 . 2 ((𝐷 𝑉 B𝐷) → ((A ↾ {𝐷})‘𝐷) = (A𝐷))
1814, 17eqtrd 2069 1 ((𝐷 𝑉 B𝐷) → ((A ∪ {⟨B, 𝐶⟩})‘𝐷) = (A𝐷))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   = wceq 1242   wcel 1390  wne 2201  cun 2909  c0 3218  {csn 3367  cop 3370  cres 4290  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-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-v 2553  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-xp 4294  df-res 4300  df-iota 4810  df-fv 4853
This theorem is referenced by:  fvpr1  5308  fvpr1g  5310  fvpr2g  5311  fvtp1g  5312  tfrlemisucaccv  5880
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