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Theorem fvsnun2 6331
Description: The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 6330. (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 5299 . . . 4 (𝐺 ↾ (𝐶 ∖ {𝐴})) = (({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ (𝐶 ∖ {𝐴}))
3 resundir 5317 . . . 4 (({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ (𝐶 ∖ {𝐴})) = (({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴})))
4 disjdif 3991 . . . . . . 7 ({𝐴} ∩ (𝐶 ∖ {𝐴})) = ∅
5 fvsnun.1 . . . . . . . . 9 𝐴 ∈ V
6 fvsnun.2 . . . . . . . . 9 𝐵 ∈ V
75, 6fnsn 5845 . . . . . . . 8 {⟨𝐴, 𝐵⟩} Fn {𝐴}
8 fnresdisj 5900 . . . . . . . 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 218 . . . . . 6 ({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) = ∅
11 residm 5336 . . . . . 6 ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴})) = (𝐹 ↾ (𝐶 ∖ {𝐴}))
1210, 11uneq12i 3726 . . . . 5 (({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴}))) = (∅ ∪ (𝐹 ↾ (𝐶 ∖ {𝐴})))
13 uncom 3718 . . . . 5 (∅ ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) = ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ ∅)
14 un0 3918 . . . . 5 ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ ∅) = (𝐹 ↾ (𝐶 ∖ {𝐴}))
1512, 13, 143eqtri 2635 . . . 4 (({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴}))) = (𝐹 ↾ (𝐶 ∖ {𝐴}))
162, 3, 153eqtri 2635 . . 3 (𝐺 ↾ (𝐶 ∖ {𝐴})) = (𝐹 ↾ (𝐶 ∖ {𝐴}))
1716fveq1i 6088 . 2 ((𝐺 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷)
18 fvres 6101 . 2 (𝐷 ∈ (𝐶 ∖ {𝐴}) → ((𝐺 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = (𝐺𝐷))
19 fvres 6101 . 2 (𝐷 ∈ (𝐶 ∖ {𝐴}) → ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = (𝐹𝐷))
2017, 18, 193eqtr3a 2667 1 (𝐷 ∈ (𝐶 ∖ {𝐴}) → (𝐺𝐷) = (𝐹𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194   = wceq 1474  wcel 1976  Vcvv 3172  cdif 3536  cun 3537  cin 3538  c0 3873  {csn 4124  cop 4130  cres 5029   Fn wfn 5784  cfv 5789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-res 5039  df-iota 5753  df-fun 5791  df-fn 5792  df-fv 5797
This theorem is referenced by:  facnn  12881
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