Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-fvsnun2 Structured version   Visualization version   GIF version

Theorem bj-fvsnun2 37257
Description: The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 7203. (Contributed by NM, 23-Sep-2007.) Put in deduction form. (Revised by BJ, 18-Mar-2023.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-fvsnun.un (𝜑𝐺 = ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
bj-fvsnun2.ex1 (𝜑𝐴𝑉)
bj-fvsnun2.ex2 (𝜑𝐵𝑊)
Assertion
Ref Expression
bj-fvsnun2 (𝜑 → (𝐺𝐴) = 𝐵)

Proof of Theorem bj-fvsnun2
StepHypRef Expression
1 bj-fvsnun.un . 2 (𝜑𝐺 = ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
2 dmres 6030 . . . . 5 dom (𝐹 ↾ (𝐶 ∖ {𝐴})) = ((𝐶 ∖ {𝐴}) ∩ dom 𝐹)
3 inss1 4237 . . . . 5 ((𝐶 ∖ {𝐴}) ∩ dom 𝐹) ⊆ (𝐶 ∖ {𝐴})
42, 3eqsstri 4030 . . . 4 dom (𝐹 ↾ (𝐶 ∖ {𝐴})) ⊆ (𝐶 ∖ {𝐴})
54a1i 11 . . 3 (𝜑 → dom (𝐹 ↾ (𝐶 ∖ {𝐴})) ⊆ (𝐶 ∖ {𝐴}))
6 neldifsnd 4793 . . 3 (𝜑 → ¬ 𝐴 ∈ (𝐶 ∖ {𝐴}))
75, 6ssneldd 3986 . 2 (𝜑 → ¬ 𝐴 ∈ dom (𝐹 ↾ (𝐶 ∖ {𝐴})))
8 bj-fvsnun2.ex1 . 2 (𝜑𝐴𝑉)
9 bj-fvsnun2.ex2 . 2 (𝜑𝐵𝑊)
101, 7, 8, 9bj-fununsn2 37255 1 (𝜑 → (𝐺𝐴) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  cdif 3948  cun 3949  cin 3950  wss 3951  {csn 4626  cop 4632  dom cdm 5685  cres 5687  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator