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Theorem fvsnun2 6942
Description: The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 6941. (Contributed by NM, 23-Sep-2007.) Put in deduction form. (Revised by BJ, 25-Feb-2023.)
Hypotheses
Ref Expression
fvsnun.1 (𝜑𝐴𝑉)
fvsnun.2 (𝜑𝐵𝑊)
fvsnun.3 𝐺 = ({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴})))
fvsnun2.4 (𝜑𝐷 ∈ (𝐶 ∖ {𝐴}))
Assertion
Ref Expression
fvsnun2 (𝜑 → (𝐺𝐷) = (𝐹𝐷))

Proof of Theorem fvsnun2
StepHypRef Expression
1 fvsnun.3 . . . . . 6 𝐺 = ({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴})))
21reseq1i 5847 . . . . 5 (𝐺 ↾ (𝐶 ∖ {𝐴})) = (({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ (𝐶 ∖ {𝐴}))
32a1i 11 . . . 4 (𝜑 → (𝐺 ↾ (𝐶 ∖ {𝐴})) = (({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ (𝐶 ∖ {𝐴})))
4 resundir 5866 . . . . 5 (({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ (𝐶 ∖ {𝐴})) = (({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴})))
54a1i 11 . . . 4 (𝜑 → (({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ (𝐶 ∖ {𝐴})) = (({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴}))))
6 disjdif 4423 . . . . . . 7 ({𝐴} ∩ (𝐶 ∖ {𝐴})) = ∅
7 fvsnun.1 . . . . . . . . 9 (𝜑𝐴𝑉)
8 fvsnun.2 . . . . . . . . 9 (𝜑𝐵𝑊)
9 fnsng 6402 . . . . . . . . 9 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} Fn {𝐴})
107, 8, 9syl2anc 584 . . . . . . . 8 (𝜑 → {⟨𝐴, 𝐵⟩} Fn {𝐴})
11 fnresdisj 6463 . . . . . . . 8 ({⟨𝐴, 𝐵⟩} Fn {𝐴} → (({𝐴} ∩ (𝐶 ∖ {𝐴})) = ∅ ↔ ({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) = ∅))
1210, 11syl 17 . . . . . . 7 (𝜑 → (({𝐴} ∩ (𝐶 ∖ {𝐴})) = ∅ ↔ ({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) = ∅))
136, 12mpbii 234 . . . . . 6 (𝜑 → ({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) = ∅)
14 residm 5884 . . . . . . 7 ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴})) = (𝐹 ↾ (𝐶 ∖ {𝐴}))
1514a1i 11 . . . . . 6 (𝜑 → ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴})) = (𝐹 ↾ (𝐶 ∖ {𝐴})))
1613, 15uneq12d 4143 . . . . 5 (𝜑 → (({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴}))) = (∅ ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))))
17 uncom 4132 . . . . . 6 (∅ ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) = ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ ∅)
1817a1i 11 . . . . 5 (𝜑 → (∅ ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) = ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ ∅))
19 un0 4347 . . . . . 6 ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ ∅) = (𝐹 ↾ (𝐶 ∖ {𝐴}))
2019a1i 11 . . . . 5 (𝜑 → ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ ∅) = (𝐹 ↾ (𝐶 ∖ {𝐴})))
2116, 18, 203eqtrd 2864 . . . 4 (𝜑 → (({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴}))) = (𝐹 ↾ (𝐶 ∖ {𝐴})))
223, 5, 213eqtrd 2864 . . 3 (𝜑 → (𝐺 ↾ (𝐶 ∖ {𝐴})) = (𝐹 ↾ (𝐶 ∖ {𝐴})))
2322fveq1d 6668 . 2 (𝜑 → ((𝐺 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷))
24 fvsnun2.4 . . 3 (𝜑𝐷 ∈ (𝐶 ∖ {𝐴}))
2524fvresd 6686 . 2 (𝜑 → ((𝐺 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = (𝐺𝐷))
2624fvresd 6686 . 2 (𝜑 → ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = (𝐹𝐷))
2723, 25, 263eqtr3d 2868 1 (𝜑 → (𝐺𝐷) = (𝐹𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1530  wcel 2107  cdif 3936  cun 3937  cin 3938  c0 4294  {csn 4563  cop 4569  cres 5555   Fn wfn 6346  cfv 6351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-res 5565  df-iota 6311  df-fun 6353  df-fn 6354  df-fv 6359
This theorem is referenced by:  facnn  13628  satfv1lem  32495
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