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Theorem fvsnun1 5303
 Description: The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 5304. (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 4551 . . . 4 (𝐺 ↾ {A}) = (({⟨A, B⟩} ∪ (𝐹 ↾ (𝐶 ∖ {A}))) ↾ {A})
3 resundir 4569 . . . . 5 (({⟨A, B⟩} ∪ (𝐹 ↾ (𝐶 ∖ {A}))) ↾ {A}) = (({⟨A, B⟩} ↾ {A}) ∪ ((𝐹 ↾ (𝐶 ∖ {A})) ↾ {A}))
4 incom 3123 . . . . . . . . 9 ((𝐶 ∖ {A}) ∩ {A}) = ({A} ∩ (𝐶 ∖ {A}))
5 disjdif 3290 . . . . . . . . 9 ({A} ∩ (𝐶 ∖ {A})) = ∅
64, 5eqtri 2057 . . . . . . . 8 ((𝐶 ∖ {A}) ∩ {A}) = ∅
7 resdisj 4694 . . . . . . . 8 (((𝐶 ∖ {A}) ∩ {A}) = ∅ → ((𝐹 ↾ (𝐶 ∖ {A})) ↾ {A}) = ∅)
86, 7ax-mp 7 . . . . . . 7 ((𝐹 ↾ (𝐶 ∖ {A})) ↾ {A}) = ∅
98uneq2i 3088 . . . . . 6 (({⟨A, B⟩} ↾ {A}) ∪ ((𝐹 ↾ (𝐶 ∖ {A})) ↾ {A})) = (({⟨A, B⟩} ↾ {A}) ∪ ∅)
10 un0 3245 . . . . . 6 (({⟨A, B⟩} ↾ {A}) ∪ ∅) = ({⟨A, B⟩} ↾ {A})
119, 10eqtri 2057 . . . . 5 (({⟨A, B⟩} ↾ {A}) ∪ ((𝐹 ↾ (𝐶 ∖ {A})) ↾ {A})) = ({⟨A, B⟩} ↾ {A})
123, 11eqtri 2057 . . . 4 (({⟨A, B⟩} ∪ (𝐹 ↾ (𝐶 ∖ {A}))) ↾ {A}) = ({⟨A, B⟩} ↾ {A})
132, 12eqtri 2057 . . 3 (𝐺 ↾ {A}) = ({⟨A, B⟩} ↾ {A})
1413fveq1i 5122 . 2 ((𝐺 ↾ {A})‘A) = (({⟨A, B⟩} ↾ {A})‘A)
15 fvsnun.1 . . . 4 A V
1615snid 3394 . . 3 A {A}
17 fvres 5141 . . 3 (A {A} → ((𝐺 ↾ {A})‘A) = (𝐺A))
1816, 17ax-mp 7 . 2 ((𝐺 ↾ {A})‘A) = (𝐺A)
19 fvres 5141 . . . 4 (A {A} → (({⟨A, B⟩} ↾ {A})‘A) = ({⟨A, B⟩}‘A))
2016, 19ax-mp 7 . . 3 (({⟨A, B⟩} ↾ {A})‘A) = ({⟨A, B⟩}‘A)
21 fvsnun.2 . . . 4 B V
2215, 21fvsn 5301 . . 3 ({⟨A, B⟩}‘A) = B
2320, 22eqtri 2057 . 2 (({⟨A, B⟩} ↾ {A})‘A) = B
2414, 18, 233eqtr3i 2065 1 (𝐺A) = B
 Colors of variables: wff set class Syntax hints:   = wceq 1242   ∈ wcel 1390  Vcvv 2551   ∖ cdif 2908   ∪ cun 2909   ∩ cin 2910  ∅c0 3218  {csn 3367  ⟨cop 3370   ↾ cres 4290  ‘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-sbc 2759  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-fv 4853 This theorem is referenced by: (None)
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