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Theorem bj-fununsn1 37236
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 6236 . . . 4 dom {⟨𝐵, 𝐶⟩} ⊆ {𝐵}
32a1i 11 . . 3 (𝜑 → dom {⟨𝐵, 𝐶⟩} ⊆ {𝐵})
4 bj-fununsn1.neq . . . 4 (𝜑 → ¬ 𝐴 = 𝐵)
5 elsni 4648 . . . 4 (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
64, 5nsyl 140 . . 3 (𝜑 → ¬ 𝐴 ∈ {𝐵})
73, 6ssneldd 3998 . 2 (𝜑 → ¬ 𝐴 ∈ dom {⟨𝐵, 𝐶⟩})
81, 7bj-funun 37235 1 (𝜑 → (𝐹𝐴) = (𝐺𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1537  wcel 2106  cun 3961  wss 3963  {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-ne 2939  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-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fv 6571
This theorem is referenced by:  bj-fvsnun1  37238  bj-fvmptunsn2  37241
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