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Theorem bj-fununsn2 37237
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 4168 . . . 4 (𝐺 ∪ {⟨𝐵, 𝐶⟩}) = ({⟨𝐵, 𝐶⟩} ∪ 𝐺)
31, 2eqtrdi 2791 . . 3 (𝜑𝐹 = ({⟨𝐵, 𝐶⟩} ∪ 𝐺))
4 bj-fununsn2.neldm . . 3 (𝜑 → ¬ 𝐵 ∈ dom 𝐺)
53, 4bj-funun 37235 . 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 2775 1 (𝜑 → (𝐹𝐵) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1537  wcel 2106  cun 3961  {csn 4631  cop 4637  dom cdm 5689  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571
This theorem is referenced by:  bj-fvsnun2  37239  bj-fvmptunsn1  37240
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