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Theorem fvsnun2 6490
Description: The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 6489. (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
fvsnun2 (𝐷 ∈ (𝐶 ∖ {𝐴}) → (𝐺𝐷) = (𝐹𝐷))

Proof of Theorem fvsnun2
StepHypRef Expression
1 fvsnun.3 . . . . 5 𝐺 = ({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴})))
21reseq1i 5424 . . . 4 (𝐺 ↾ (𝐶 ∖ {𝐴})) = (({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ (𝐶 ∖ {𝐴}))
3 resundir 5446 . . . 4 (({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ (𝐶 ∖ {𝐴})) = (({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴})))
4 disjdif 4073 . . . . . . 7 ({𝐴} ∩ (𝐶 ∖ {𝐴})) = ∅
5 fvsnun.1 . . . . . . . . 9 𝐴 ∈ V
6 fvsnun.2 . . . . . . . . 9 𝐵 ∈ V
75, 6fnsn 5984 . . . . . . . 8 {⟨𝐴, 𝐵⟩} Fn {𝐴}
8 fnresdisj 6039 . . . . . . . 8 ({⟨𝐴, 𝐵⟩} Fn {𝐴} → (({𝐴} ∩ (𝐶 ∖ {𝐴})) = ∅ ↔ ({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) = ∅))
97, 8ax-mp 5 . . . . . . 7 (({𝐴} ∩ (𝐶 ∖ {𝐴})) = ∅ ↔ ({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) = ∅)
104, 9mpbi 220 . . . . . 6 ({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) = ∅
11 residm 5465 . . . . . 6 ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴})) = (𝐹 ↾ (𝐶 ∖ {𝐴}))
1210, 11uneq12i 3798 . . . . 5 (({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴}))) = (∅ ∪ (𝐹 ↾ (𝐶 ∖ {𝐴})))
13 uncom 3790 . . . . 5 (∅ ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) = ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ ∅)
14 un0 4000 . . . . 5 ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ ∅) = (𝐹 ↾ (𝐶 ∖ {𝐴}))
1512, 13, 143eqtri 2677 . . . 4 (({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴}))) = (𝐹 ↾ (𝐶 ∖ {𝐴}))
162, 3, 153eqtri 2677 . . 3 (𝐺 ↾ (𝐶 ∖ {𝐴})) = (𝐹 ↾ (𝐶 ∖ {𝐴}))
1716fveq1i 6230 . 2 ((𝐺 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷)
18 fvres 6245 . 2 (𝐷 ∈ (𝐶 ∖ {𝐴}) → ((𝐺 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = (𝐺𝐷))
19 fvres 6245 . 2 (𝐷 ∈ (𝐶 ∖ {𝐴}) → ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = (𝐹𝐷))
2017, 18, 193eqtr3a 2709 1 (𝐷 ∈ (𝐶 ∖ {𝐴}) → (𝐺𝐷) = (𝐹𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1523  wcel 2030  Vcvv 3231  cdif 3604  cun 3605  cin 3606  c0 3948  {csn 4210  cop 4216  cres 5145   Fn wfn 5921  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-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  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-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-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-res 5155  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934
This theorem is referenced by:  facnn  13102
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