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Theorem fvunsn 7177
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 5997 . . . 4 ((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷}) = ((𝐴 ↾ {𝐷}) ∪ ({⟨𝐵, 𝐶⟩} ↾ {𝐷}))
2 nelsn 4669 . . . . . . 7 (𝐵𝐷 → ¬ 𝐵 ∈ {𝐷})
3 ressnop0 7151 . . . . . . 7 𝐵 ∈ {𝐷} → ({⟨𝐵, 𝐶⟩} ↾ {𝐷}) = ∅)
42, 3syl 17 . . . . . 6 (𝐵𝐷 → ({⟨𝐵, 𝐶⟩} ↾ {𝐷}) = ∅)
54uneq2d 4164 . . . . 5 (𝐵𝐷 → ((𝐴 ↾ {𝐷}) ∪ ({⟨𝐵, 𝐶⟩} ↾ {𝐷})) = ((𝐴 ↾ {𝐷}) ∪ ∅))
6 un0 4391 . . . . 5 ((𝐴 ↾ {𝐷}) ∪ ∅) = (𝐴 ↾ {𝐷})
75, 6eqtrdi 2789 . . . 4 (𝐵𝐷 → ((𝐴 ↾ {𝐷}) ∪ ({⟨𝐵, 𝐶⟩} ↾ {𝐷})) = (𝐴 ↾ {𝐷}))
81, 7eqtrid 2785 . . 3 (𝐵𝐷 → ((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷}) = (𝐴 ↾ {𝐷}))
98fveq1d 6894 . 2 (𝐵𝐷 → (((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷})‘𝐷) = ((𝐴 ↾ {𝐷})‘𝐷))
10 fvressn 7160 . . 3 (𝐷 ∈ V → (((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷})‘𝐷) = ((𝐴 ∪ {⟨𝐵, 𝐶⟩})‘𝐷))
11 fvprc 6884 . . . 4 𝐷 ∈ V → (((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷})‘𝐷) = ∅)
12 fvprc 6884 . . . 4 𝐷 ∈ V → ((𝐴 ∪ {⟨𝐵, 𝐶⟩})‘𝐷) = ∅)
1311, 12eqtr4d 2776 . . 3 𝐷 ∈ V → (((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷})‘𝐷) = ((𝐴 ∪ {⟨𝐵, 𝐶⟩})‘𝐷))
1410, 13pm2.61i 182 . 2 (((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷})‘𝐷) = ((𝐴 ∪ {⟨𝐵, 𝐶⟩})‘𝐷)
15 fvressn 7160 . . 3 (𝐷 ∈ V → ((𝐴 ↾ {𝐷})‘𝐷) = (𝐴𝐷))
16 fvprc 6884 . . . 4 𝐷 ∈ V → ((𝐴 ↾ {𝐷})‘𝐷) = ∅)
17 fvprc 6884 . . . 4 𝐷 ∈ V → (𝐴𝐷) = ∅)
1816, 17eqtr4d 2776 . . 3 𝐷 ∈ V → ((𝐴 ↾ {𝐷})‘𝐷) = (𝐴𝐷))
1915, 18pm2.61i 182 . 2 ((𝐴 ↾ {𝐷})‘𝐷) = (𝐴𝐷)
209, 14, 193eqtr3g 2796 1 (𝐵𝐷 → ((𝐴 ∪ {⟨𝐵, 𝐶⟩})‘𝐷) = (𝐴𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1542  wcel 2107  wne 2941  Vcvv 3475  cun 3947  c0 4323  {csn 4629  cop 4635  cres 5679  cfv 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-res 5689  df-iota 6496  df-fv 6552
This theorem is referenced by:  fvpr1g  7188  fvpr2gOLD  7190  fvpr1OLD  7192  fvtp1  7196  fvtp1g  7199  ac6sfi  9287  cats1un  14671  ruclem6  16178  ruclem7  16179  wlkp1lem5  28934  wlkp1lem6  28935  fnchoice  43713  nnsum4primeseven  46468  nnsum4primesevenALTV  46469
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