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 34540
 Description: The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 6947. (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 5877 . . . . 5 dom (𝐹 ↾ (𝐶 ∖ {𝐴})) = ((𝐶 ∖ {𝐴}) ∩ dom 𝐹)
3 inss1 4207 . . . . 5 ((𝐶 ∖ {𝐴}) ∩ dom 𝐹) ⊆ (𝐶 ∖ {𝐴})
42, 3eqsstri 4003 . . . 4 dom (𝐹 ↾ (𝐶 ∖ {𝐴})) ⊆ (𝐶 ∖ {𝐴})
54a1i 11 . . 3 (𝜑 → dom (𝐹 ↾ (𝐶 ∖ {𝐴})) ⊆ (𝐶 ∖ {𝐴}))
6 neldifsnd 4728 . . 3 (𝜑 → ¬ 𝐴 ∈ (𝐶 ∖ {𝐴}))
75, 6ssneldd 3972 . 2 (𝜑 → ¬ 𝐴 ∈ dom (𝐹 ↾ (𝐶 ∖ {𝐴})))
8 bj-fvsnun2.ex1 . 2 (𝜑𝐴𝑉)
9 bj-fvsnun2.ex2 . 2 (𝜑𝐵𝑊)
101, 7, 8, 9bj-fununsn2 34538 1 (𝜑 → (𝐺𝐴) = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114   ∖ cdif 3935   ∪ cun 3936   ∩ cin 3937   ⊆ wss 3938  {csn 4569  ⟨cop 4575  dom cdm 5557   ↾ cres 5559  ‘cfv 6357 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fv 6365 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator