Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-fununsn2 Structured version   Visualization version   GIF version

Theorem bj-fununsn2 37255
Description: Value of a function expressed as a union of a function and a singleton on a couple (with disjoint domain) at the first component of that couple. (Contributed by BJ, 18-Mar-2023.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-fununsn.un (𝜑𝐹 = (𝐺 ∪ {⟨𝐵, 𝐶⟩}))
bj-fununsn2.neldm (𝜑 → ¬ 𝐵 ∈ dom 𝐺)
bj-fununsn2.ex1 (𝜑𝐵𝑉)
bj-fununsn2.ex2 (𝜑𝐶𝑊)
Assertion
Ref Expression
bj-fununsn2 (𝜑 → (𝐹𝐵) = 𝐶)

Proof of Theorem bj-fununsn2
StepHypRef Expression
1 bj-fununsn.un . . . 4 (𝜑𝐹 = (𝐺 ∪ {⟨𝐵, 𝐶⟩}))
2 uncom 4158 . . . 4 (𝐺 ∪ {⟨𝐵, 𝐶⟩}) = ({⟨𝐵, 𝐶⟩} ∪ 𝐺)
31, 2eqtrdi 2793 . . 3 (𝜑𝐹 = ({⟨𝐵, 𝐶⟩} ∪ 𝐺))
4 bj-fununsn2.neldm . . 3 (𝜑 → ¬ 𝐵 ∈ dom 𝐺)
53, 4bj-funun 37253 . 2 (𝜑 → (𝐹𝐵) = ({⟨𝐵, 𝐶⟩}‘𝐵))
6 bj-fununsn2.ex1 . . 3 (𝜑𝐵𝑉)
7 bj-fununsn2.ex2 . . 3 (𝜑𝐶𝑊)
8 fvsng 7200 . . 3 ((𝐵𝑉𝐶𝑊) → ({⟨𝐵, 𝐶⟩}‘𝐵) = 𝐶)
96, 7, 8syl2anc 584 . 2 (𝜑 → ({⟨𝐵, 𝐶⟩}‘𝐵) = 𝐶)
105, 9eqtrd 2777 1 (𝜑 → (𝐹𝐵) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1540  wcel 2108  cun 3949  {csn 4626  cop 4632  dom cdm 5685  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-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:  bj-fvsnun2  37257  bj-fvmptunsn1  37258
  Copyright terms: Public domain W3C validator