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Theorem fvunsn 7198
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 6014 . . . 4 ((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷}) = ((𝐴 ↾ {𝐷}) ∪ ({⟨𝐵, 𝐶⟩} ↾ {𝐷}))
2 nelsn 4670 . . . . . . 7 (𝐵𝐷 → ¬ 𝐵 ∈ {𝐷})
3 ressnop0 7172 . . . . . . 7 𝐵 ∈ {𝐷} → ({⟨𝐵, 𝐶⟩} ↾ {𝐷}) = ∅)
42, 3syl 17 . . . . . 6 (𝐵𝐷 → ({⟨𝐵, 𝐶⟩} ↾ {𝐷}) = ∅)
54uneq2d 4177 . . . . 5 (𝐵𝐷 → ((𝐴 ↾ {𝐷}) ∪ ({⟨𝐵, 𝐶⟩} ↾ {𝐷})) = ((𝐴 ↾ {𝐷}) ∪ ∅))
6 un0 4399 . . . . 5 ((𝐴 ↾ {𝐷}) ∪ ∅) = (𝐴 ↾ {𝐷})
75, 6eqtrdi 2790 . . . 4 (𝐵𝐷 → ((𝐴 ↾ {𝐷}) ∪ ({⟨𝐵, 𝐶⟩} ↾ {𝐷})) = (𝐴 ↾ {𝐷}))
81, 7eqtrid 2786 . . 3 (𝐵𝐷 → ((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷}) = (𝐴 ↾ {𝐷}))
98fveq1d 6908 . 2 (𝐵𝐷 → (((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷})‘𝐷) = ((𝐴 ↾ {𝐷})‘𝐷))
10 fvressn 7181 . . 3 (𝐷 ∈ V → (((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷})‘𝐷) = ((𝐴 ∪ {⟨𝐵, 𝐶⟩})‘𝐷))
11 fvprc 6898 . . . 4 𝐷 ∈ V → (((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷})‘𝐷) = ∅)
12 fvprc 6898 . . . 4 𝐷 ∈ V → ((𝐴 ∪ {⟨𝐵, 𝐶⟩})‘𝐷) = ∅)
1311, 12eqtr4d 2777 . . 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 7181 . . 3 (𝐷 ∈ V → ((𝐴 ↾ {𝐷})‘𝐷) = (𝐴𝐷))
16 fvprc 6898 . . . 4 𝐷 ∈ V → ((𝐴 ↾ {𝐷})‘𝐷) = ∅)
17 fvprc 6898 . . . 4 𝐷 ∈ V → (𝐴𝐷) = ∅)
1816, 17eqtr4d 2777 . . 3 𝐷 ∈ V → ((𝐴 ↾ {𝐷})‘𝐷) = (𝐴𝐷))
1915, 18pm2.61i 182 . 2 ((𝐴 ↾ {𝐷})‘𝐷) = (𝐴𝐷)
209, 14, 193eqtr3g 2797 1 (𝐵𝐷 → ((𝐴 ∪ {⟨𝐵, 𝐶⟩})‘𝐷) = (𝐴𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1536  wcel 2105  wne 2937  Vcvv 3477  cun 3960  c0 4338  {csn 4630  cop 4636  cres 5690  cfv 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-xp 5694  df-res 5700  df-iota 6515  df-fv 6570
This theorem is referenced by:  fvpr1g  7209  fvtp1  7214  fvtp1g  7217  f1ounsn  7291  ac6sfi  9317  cats1un  14755  ruclem6  16267  ruclem7  16268  wlkp1lem5  29709  wlkp1lem6  29710  fnchoice  44966  nnsum4primeseven  47724  nnsum4primesevenALTV  47725
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