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Theorem fvunsn 6486
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 5446 . . . 4 ((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷}) = ((𝐴 ↾ {𝐷}) ∪ ({⟨𝐵, 𝐶⟩} ↾ {𝐷}))
2 nelsn 4245 . . . . . . 7 (𝐵𝐷 → ¬ 𝐵 ∈ {𝐷})
3 ressnop0 6460 . . . . . . 7 𝐵 ∈ {𝐷} → ({⟨𝐵, 𝐶⟩} ↾ {𝐷}) = ∅)
42, 3syl 17 . . . . . 6 (𝐵𝐷 → ({⟨𝐵, 𝐶⟩} ↾ {𝐷}) = ∅)
54uneq2d 3800 . . . . 5 (𝐵𝐷 → ((𝐴 ↾ {𝐷}) ∪ ({⟨𝐵, 𝐶⟩} ↾ {𝐷})) = ((𝐴 ↾ {𝐷}) ∪ ∅))
6 un0 4000 . . . . 5 ((𝐴 ↾ {𝐷}) ∪ ∅) = (𝐴 ↾ {𝐷})
75, 6syl6eq 2701 . . . 4 (𝐵𝐷 → ((𝐴 ↾ {𝐷}) ∪ ({⟨𝐵, 𝐶⟩} ↾ {𝐷})) = (𝐴 ↾ {𝐷}))
81, 7syl5eq 2697 . . 3 (𝐵𝐷 → ((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷}) = (𝐴 ↾ {𝐷}))
98fveq1d 6231 . 2 (𝐵𝐷 → (((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷})‘𝐷) = ((𝐴 ↾ {𝐷})‘𝐷))
10 fvressn 6469 . . 3 (𝐷 ∈ V → (((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷})‘𝐷) = ((𝐴 ∪ {⟨𝐵, 𝐶⟩})‘𝐷))
11 fvprc 6223 . . . 4 𝐷 ∈ V → (((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷})‘𝐷) = ∅)
12 fvprc 6223 . . . 4 𝐷 ∈ V → ((𝐴 ∪ {⟨𝐵, 𝐶⟩})‘𝐷) = ∅)
1311, 12eqtr4d 2688 . . 3 𝐷 ∈ V → (((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷})‘𝐷) = ((𝐴 ∪ {⟨𝐵, 𝐶⟩})‘𝐷))
1410, 13pm2.61i 176 . 2 (((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷})‘𝐷) = ((𝐴 ∪ {⟨𝐵, 𝐶⟩})‘𝐷)
15 fvressn 6469 . . 3 (𝐷 ∈ V → ((𝐴 ↾ {𝐷})‘𝐷) = (𝐴𝐷))
16 fvprc 6223 . . . 4 𝐷 ∈ V → ((𝐴 ↾ {𝐷})‘𝐷) = ∅)
17 fvprc 6223 . . . 4 𝐷 ∈ V → (𝐴𝐷) = ∅)
1816, 17eqtr4d 2688 . . 3 𝐷 ∈ V → ((𝐴 ↾ {𝐷})‘𝐷) = (𝐴𝐷))
1915, 18pm2.61i 176 . 2 ((𝐴 ↾ {𝐷})‘𝐷) = (𝐴𝐷)
209, 14, 193eqtr3g 2708 1 (𝐵𝐷 → ((𝐴 ∪ {⟨𝐵, 𝐶⟩})‘𝐷) = (𝐴𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1523  wcel 2030  wne 2823  Vcvv 3231  cun 3605  c0 3948  {csn 4210  cop 4216  cres 5145  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-res 5155  df-iota 5889  df-fv 5934
This theorem is referenced by:  fvpr1  6497  fvpr1g  6499  fvpr2g  6500  fvtp1  6501  fvtp1g  6504  ac6sfi  8245  cats1un  13521  ruclem6  15008  ruclem7  15009  wlkp1lem5  26630  wlkp1lem6  26631  fnchoice  39502  nnsum4primeseven  42013  nnsum4primesevenALTV  42014
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