MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvunsn Structured version   Visualization version   GIF version

Theorem fvunsn 7126
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 5953 . . . 4 ((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷}) = ((𝐴 ↾ {𝐷}) ∪ ({⟨𝐵, 𝐶⟩} ↾ {𝐷}))
2 nelsn 4627 . . . . . . 7 (𝐵𝐷 → ¬ 𝐵 ∈ {𝐷})
3 ressnop0 7100 . . . . . . 7 𝐵 ∈ {𝐷} → ({⟨𝐵, 𝐶⟩} ↾ {𝐷}) = ∅)
42, 3syl 17 . . . . . 6 (𝐵𝐷 → ({⟨𝐵, 𝐶⟩} ↾ {𝐷}) = ∅)
54uneq2d 4124 . . . . 5 (𝐵𝐷 → ((𝐴 ↾ {𝐷}) ∪ ({⟨𝐵, 𝐶⟩} ↾ {𝐷})) = ((𝐴 ↾ {𝐷}) ∪ ∅))
6 un0 4351 . . . . 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 6845 . 2 (𝐵𝐷 → (((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷})‘𝐷) = ((𝐴 ↾ {𝐷})‘𝐷))
10 fvressn 7109 . . 3 (𝐷 ∈ V → (((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷})‘𝐷) = ((𝐴 ∪ {⟨𝐵, 𝐶⟩})‘𝐷))
11 fvprc 6835 . . . 4 𝐷 ∈ V → (((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷})‘𝐷) = ∅)
12 fvprc 6835 . . . 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 7109 . . 3 (𝐷 ∈ V → ((𝐴 ↾ {𝐷})‘𝐷) = (𝐴𝐷))
16 fvprc 6835 . . . 4 𝐷 ∈ V → ((𝐴 ↾ {𝐷})‘𝐷) = ∅)
17 fvprc 6835 . . . 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 2940  Vcvv 3444  cun 3909  c0 4283  {csn 4587  cop 4593  cres 5636  cfv 6497
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 5257  ax-nul 5264  ax-pr 5385
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 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-xp 5640  df-res 5646  df-iota 6449  df-fv 6505
This theorem is referenced by:  fvpr1g  7137  fvpr2gOLD  7139  fvpr1OLD  7141  fvtp1  7145  fvtp1g  7148  ac6sfi  9234  cats1un  14615  ruclem6  16122  ruclem7  16123  wlkp1lem5  28667  wlkp1lem6  28668  fnchoice  43322  nnsum4primeseven  46078  nnsum4primesevenALTV  46079
  Copyright terms: Public domain W3C validator