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Theorem bj-fununsn1 37254
Description: Value of a function expressed as a union of a function and a singleton on a couple (with disjoint domain) at a point not equal to 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-fununsn1.neq (𝜑 → ¬ 𝐴 = 𝐵)
Assertion
Ref Expression
bj-fununsn1 (𝜑 → (𝐹𝐴) = (𝐺𝐴))

Proof of Theorem bj-fununsn1
StepHypRef Expression
1 bj-fununsn.un . 2 (𝜑𝐹 = (𝐺 ∪ {⟨𝐵, 𝐶⟩}))
2 dmsnopss 6234 . . . 4 dom {⟨𝐵, 𝐶⟩} ⊆ {𝐵}
32a1i 11 . . 3 (𝜑 → dom {⟨𝐵, 𝐶⟩} ⊆ {𝐵})
4 bj-fununsn1.neq . . . 4 (𝜑 → ¬ 𝐴 = 𝐵)
5 elsni 4643 . . . 4 (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
64, 5nsyl 140 . . 3 (𝜑 → ¬ 𝐴 ∈ {𝐵})
73, 6ssneldd 3986 . 2 (𝜑 → ¬ 𝐴 ∈ dom {⟨𝐵, 𝐶⟩})
81, 7bj-funun 37253 1 (𝜑 → (𝐹𝐴) = (𝐺𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1540  wcel 2108  cun 3949  wss 3951  {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-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-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fv 6569
This theorem is referenced by:  bj-fvsnun1  37256  bj-fvmptunsn2  37259
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