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Theorem fvunsn 6922
Description: Remove an ordered pair not participating in a function value. (Contributed by NM, 1-Oct-2013.) (Revised by Mario Carneiro, 28-May-2014.)
Assertion
Ref Expression
fvunsn (𝐵𝐷 → ((𝐴 ∪ {⟨𝐵, 𝐶⟩})‘𝐷) = (𝐴𝐷))

Proof of Theorem fvunsn
StepHypRef Expression
1 resundir 5837 . . . 4 ((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷}) = ((𝐴 ↾ {𝐷}) ∪ ({⟨𝐵, 𝐶⟩} ↾ {𝐷}))
2 nelsn 4568 . . . . . . 7 (𝐵𝐷 → ¬ 𝐵 ∈ {𝐷})
3 ressnop0 6896 . . . . . . 7 𝐵 ∈ {𝐷} → ({⟨𝐵, 𝐶⟩} ↾ {𝐷}) = ∅)
42, 3syl 17 . . . . . 6 (𝐵𝐷 → ({⟨𝐵, 𝐶⟩} ↾ {𝐷}) = ∅)
54uneq2d 4093 . . . . 5 (𝐵𝐷 → ((𝐴 ↾ {𝐷}) ∪ ({⟨𝐵, 𝐶⟩} ↾ {𝐷})) = ((𝐴 ↾ {𝐷}) ∪ ∅))
6 un0 4301 . . . . 5 ((𝐴 ↾ {𝐷}) ∪ ∅) = (𝐴 ↾ {𝐷})
75, 6eqtrdi 2852 . . . 4 (𝐵𝐷 → ((𝐴 ↾ {𝐷}) ∪ ({⟨𝐵, 𝐶⟩} ↾ {𝐷})) = (𝐴 ↾ {𝐷}))
81, 7syl5eq 2848 . . 3 (𝐵𝐷 → ((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷}) = (𝐴 ↾ {𝐷}))
98fveq1d 6651 . 2 (𝐵𝐷 → (((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷})‘𝐷) = ((𝐴 ↾ {𝐷})‘𝐷))
10 fvressn 6905 . . 3 (𝐷 ∈ V → (((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷})‘𝐷) = ((𝐴 ∪ {⟨𝐵, 𝐶⟩})‘𝐷))
11 fvprc 6642 . . . 4 𝐷 ∈ V → (((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷})‘𝐷) = ∅)
12 fvprc 6642 . . . 4 𝐷 ∈ V → ((𝐴 ∪ {⟨𝐵, 𝐶⟩})‘𝐷) = ∅)
1311, 12eqtr4d 2839 . . 3 𝐷 ∈ V → (((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷})‘𝐷) = ((𝐴 ∪ {⟨𝐵, 𝐶⟩})‘𝐷))
1410, 13pm2.61i 185 . 2 (((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷})‘𝐷) = ((𝐴 ∪ {⟨𝐵, 𝐶⟩})‘𝐷)
15 fvressn 6905 . . 3 (𝐷 ∈ V → ((𝐴 ↾ {𝐷})‘𝐷) = (𝐴𝐷))
16 fvprc 6642 . . . 4 𝐷 ∈ V → ((𝐴 ↾ {𝐷})‘𝐷) = ∅)
17 fvprc 6642 . . . 4 𝐷 ∈ V → (𝐴𝐷) = ∅)
1816, 17eqtr4d 2839 . . 3 𝐷 ∈ V → ((𝐴 ↾ {𝐷})‘𝐷) = (𝐴𝐷))
1915, 18pm2.61i 185 . 2 ((𝐴 ↾ {𝐷})‘𝐷) = (𝐴𝐷)
209, 14, 193eqtr3g 2859 1 (𝐵𝐷 → ((𝐴 ∪ {⟨𝐵, 𝐶⟩})‘𝐷) = (𝐴𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1538  wcel 2112  wne 2990  Vcvv 3444  cun 3882  c0 4246  {csn 4528  cop 4534  cres 5525  cfv 6328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-xp 5529  df-res 5535  df-iota 6287  df-fv 6336
This theorem is referenced by:  fvpr1  6933  fvpr1g  6935  fvpr2g  6936  fvtp1  6938  fvtp1g  6941  ac6sfi  8750  cats1un  14078  ruclem6  15584  ruclem7  15585  wlkp1lem5  27471  wlkp1lem6  27472  fnchoice  41655  nnsum4primeseven  44315  nnsum4primesevenALTV  44316
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