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Theorem fvsnun1 6331
Description: The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 6332. (Contributed by NM, 23-Sep-2007.)
Hypotheses
Ref Expression
fvsnun.1 𝐴 ∈ V
fvsnun.2 𝐵 ∈ V
fvsnun.3 𝐺 = ({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴})))
Assertion
Ref Expression
fvsnun1 (𝐺𝐴) = 𝐵

Proof of Theorem fvsnun1
StepHypRef Expression
1 fvsnun.3 . . . . 5 𝐺 = ({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴})))
21reseq1i 5300 . . . 4 (𝐺 ↾ {𝐴}) = (({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ {𝐴})
3 resundir 5318 . . . . 5 (({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ {𝐴}) = (({⟨𝐴, 𝐵⟩} ↾ {𝐴}) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴}))
4 incom 3767 . . . . . . . . 9 ((𝐶 ∖ {𝐴}) ∩ {𝐴}) = ({𝐴} ∩ (𝐶 ∖ {𝐴}))
5 disjdif 3992 . . . . . . . . 9 ({𝐴} ∩ (𝐶 ∖ {𝐴})) = ∅
64, 5eqtri 2632 . . . . . . . 8 ((𝐶 ∖ {𝐴}) ∩ {𝐴}) = ∅
7 resdisj 5468 . . . . . . . 8 (((𝐶 ∖ {𝐴}) ∩ {𝐴}) = ∅ → ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴}) = ∅)
86, 7ax-mp 5 . . . . . . 7 ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴}) = ∅
98uneq2i 3726 . . . . . 6 (({⟨𝐴, 𝐵⟩} ↾ {𝐴}) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴})) = (({⟨𝐴, 𝐵⟩} ↾ {𝐴}) ∪ ∅)
10 un0 3919 . . . . . 6 (({⟨𝐴, 𝐵⟩} ↾ {𝐴}) ∪ ∅) = ({⟨𝐴, 𝐵⟩} ↾ {𝐴})
119, 10eqtri 2632 . . . . 5 (({⟨𝐴, 𝐵⟩} ↾ {𝐴}) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴})) = ({⟨𝐴, 𝐵⟩} ↾ {𝐴})
123, 11eqtri 2632 . . . 4 (({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ {𝐴}) = ({⟨𝐴, 𝐵⟩} ↾ {𝐴})
132, 12eqtri 2632 . . 3 (𝐺 ↾ {𝐴}) = ({⟨𝐴, 𝐵⟩} ↾ {𝐴})
1413fveq1i 6089 . 2 ((𝐺 ↾ {𝐴})‘𝐴) = (({⟨𝐴, 𝐵⟩} ↾ {𝐴})‘𝐴)
15 fvsnun.1 . . . 4 𝐴 ∈ V
1615snid 4155 . . 3 𝐴 ∈ {𝐴}
17 fvres 6102 . . 3 (𝐴 ∈ {𝐴} → ((𝐺 ↾ {𝐴})‘𝐴) = (𝐺𝐴))
1816, 17ax-mp 5 . 2 ((𝐺 ↾ {𝐴})‘𝐴) = (𝐺𝐴)
19 fvres 6102 . . . 4 (𝐴 ∈ {𝐴} → (({⟨𝐴, 𝐵⟩} ↾ {𝐴})‘𝐴) = ({⟨𝐴, 𝐵⟩}‘𝐴))
2016, 19ax-mp 5 . . 3 (({⟨𝐴, 𝐵⟩} ↾ {𝐴})‘𝐴) = ({⟨𝐴, 𝐵⟩}‘𝐴)
21 fvsnun.2 . . . 4 𝐵 ∈ V
2215, 21fvsn 6329 . . 3 ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵
2320, 22eqtri 2632 . 2 (({⟨𝐴, 𝐵⟩} ↾ {𝐴})‘𝐴) = 𝐵
2414, 18, 233eqtr3i 2640 1 (𝐺𝐴) = 𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1475  wcel 1977  Vcvv 3173  cdif 3537  cun 3538  cin 3539  c0 3874  {csn 4125  cop 4131  cres 5030  cfv 5790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4704  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-br 4579  df-opab 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-res 5040  df-iota 5754  df-fun 5792  df-fv 5798
This theorem is referenced by:  fac0  12883  ruclem4  14751
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