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Theorem fvsnun1 7203
Description: The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 7204. (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})))
Assertion
Ref Expression
fvsnun1 (𝜑 → (𝐺𝐴) = 𝐵)

Proof of Theorem fvsnun1
StepHypRef Expression
1 fvsnun.3 . . . . 5 𝐺 = ({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴})))
21reseq1i 5992 . . . 4 (𝐺 ↾ {𝐴}) = (({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ {𝐴})
3 resundir 6011 . . . . 5 (({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ {𝐴}) = (({⟨𝐴, 𝐵⟩} ↾ {𝐴}) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴}))
4 disjdifr 4472 . . . . . . . 8 ((𝐶 ∖ {𝐴}) ∩ {𝐴}) = ∅
5 resdisj 6188 . . . . . . . 8 (((𝐶 ∖ {𝐴}) ∩ {𝐴}) = ∅ → ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴}) = ∅)
64, 5ax-mp 5 . . . . . . 7 ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴}) = ∅
76uneq2i 4164 . . . . . 6 (({⟨𝐴, 𝐵⟩} ↾ {𝐴}) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴})) = (({⟨𝐴, 𝐵⟩} ↾ {𝐴}) ∪ ∅)
8 un0 4393 . . . . . 6 (({⟨𝐴, 𝐵⟩} ↾ {𝐴}) ∪ ∅) = ({⟨𝐴, 𝐵⟩} ↾ {𝐴})
97, 8eqtri 2764 . . . . 5 (({⟨𝐴, 𝐵⟩} ↾ {𝐴}) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴})) = ({⟨𝐴, 𝐵⟩} ↾ {𝐴})
103, 9eqtri 2764 . . . 4 (({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ {𝐴}) = ({⟨𝐴, 𝐵⟩} ↾ {𝐴})
112, 10eqtri 2764 . . 3 (𝐺 ↾ {𝐴}) = ({⟨𝐴, 𝐵⟩} ↾ {𝐴})
1211fveq1i 6906 . 2 ((𝐺 ↾ {𝐴})‘𝐴) = (({⟨𝐴, 𝐵⟩} ↾ {𝐴})‘𝐴)
13 fvsnun.1 . . . 4 (𝜑𝐴𝑉)
14 snidg 4659 . . . 4 (𝐴𝑉𝐴 ∈ {𝐴})
1513, 14syl 17 . . 3 (𝜑𝐴 ∈ {𝐴})
1615fvresd 6925 . 2 (𝜑 → ((𝐺 ↾ {𝐴})‘𝐴) = (𝐺𝐴))
1715fvresd 6925 . . 3 (𝜑 → (({⟨𝐴, 𝐵⟩} ↾ {𝐴})‘𝐴) = ({⟨𝐴, 𝐵⟩}‘𝐴))
18 fvsnun.2 . . . 4 (𝜑𝐵𝑊)
19 fvsng 7201 . . . 4 ((𝐴𝑉𝐵𝑊) → ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵)
2013, 18, 19syl2anc 584 . . 3 (𝜑 → ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵)
2117, 20eqtrd 2776 . 2 (𝜑 → (({⟨𝐴, 𝐵⟩} ↾ {𝐴})‘𝐴) = 𝐵)
2212, 16, 213eqtr3a 2800 1 (𝜑 → (𝐺𝐴) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  cdif 3947  cun 3948  cin 3949  c0 4332  {csn 4625  cop 4631  cres 5686  cfv 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-res 5696  df-iota 6513  df-fun 6562  df-fv 6568
This theorem is referenced by:  fac0  14316  ruclem4  16271  satfv1lem  35368
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