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Theorem fvsnun2 5304
Description: The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 5303. (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 4551 . . . 4 (𝐺 ↾ (𝐶 ∖ {A})) = (({⟨A, B⟩} ∪ (𝐹 ↾ (𝐶 ∖ {A}))) ↾ (𝐶 ∖ {A}))
3 resundir 4569 . . . 4 (({⟨A, B⟩} ∪ (𝐹 ↾ (𝐶 ∖ {A}))) ↾ (𝐶 ∖ {A})) = (({⟨A, B⟩} ↾ (𝐶 ∖ {A})) ∪ ((𝐹 ↾ (𝐶 ∖ {A})) ↾ (𝐶 ∖ {A})))
4 disjdif 3290 . . . . . . 7 ({A} ∩ (𝐶 ∖ {A})) = ∅
5 fvsnun.1 . . . . . . . . 9 A V
6 fvsnun.2 . . . . . . . . 9 B V
75, 6fnsn 4896 . . . . . . . 8 {⟨A, B⟩} Fn {A}
8 fnresdisj 4952 . . . . . . . 8 ({⟨A, B⟩} Fn {A} → (({A} ∩ (𝐶 ∖ {A})) = ∅ ↔ ({⟨A, B⟩} ↾ (𝐶 ∖ {A})) = ∅))
97, 8ax-mp 7 . . . . . . 7 (({A} ∩ (𝐶 ∖ {A})) = ∅ ↔ ({⟨A, B⟩} ↾ (𝐶 ∖ {A})) = ∅)
104, 9mpbi 133 . . . . . 6 ({⟨A, B⟩} ↾ (𝐶 ∖ {A})) = ∅
11 residm 4585 . . . . . 6 ((𝐹 ↾ (𝐶 ∖ {A})) ↾ (𝐶 ∖ {A})) = (𝐹 ↾ (𝐶 ∖ {A}))
1210, 11uneq12i 3089 . . . . 5 (({⟨A, B⟩} ↾ (𝐶 ∖ {A})) ∪ ((𝐹 ↾ (𝐶 ∖ {A})) ↾ (𝐶 ∖ {A}))) = (∅ ∪ (𝐹 ↾ (𝐶 ∖ {A})))
13 uncom 3081 . . . . 5 (∅ ∪ (𝐹 ↾ (𝐶 ∖ {A}))) = ((𝐹 ↾ (𝐶 ∖ {A})) ∪ ∅)
14 un0 3245 . . . . 5 ((𝐹 ↾ (𝐶 ∖ {A})) ∪ ∅) = (𝐹 ↾ (𝐶 ∖ {A}))
1512, 13, 143eqtri 2061 . . . 4 (({⟨A, B⟩} ↾ (𝐶 ∖ {A})) ∪ ((𝐹 ↾ (𝐶 ∖ {A})) ↾ (𝐶 ∖ {A}))) = (𝐹 ↾ (𝐶 ∖ {A}))
162, 3, 153eqtri 2061 . . 3 (𝐺 ↾ (𝐶 ∖ {A})) = (𝐹 ↾ (𝐶 ∖ {A}))
1716fveq1i 5122 . 2 ((𝐺 ↾ (𝐶 ∖ {A}))‘𝐷) = ((𝐹 ↾ (𝐶 ∖ {A}))‘𝐷)
18 fvres 5141 . 2 (𝐷 (𝐶 ∖ {A}) → ((𝐺 ↾ (𝐶 ∖ {A}))‘𝐷) = (𝐺𝐷))
19 fvres 5141 . 2 (𝐷 (𝐶 ∖ {A}) → ((𝐹 ↾ (𝐶 ∖ {A}))‘𝐷) = (𝐹𝐷))
2017, 18, 193eqtr3a 2093 1 (𝐷 (𝐶 ∖ {A}) → (𝐺𝐷) = (𝐹𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242   wcel 1390  Vcvv 2551  cdif 2908  cun 2909  cin 2910  c0 3218  {csn 3367  cop 3370  cres 4290   Fn wfn 4840  cfv 4845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-res 4300  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853
This theorem is referenced by: (None)
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