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Theorem bj-fvmptunsn1 37240
Description: Value of a function expressed as a union of a mapsto expression 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-fvmptunsn.un (𝜑𝐹 = ((𝑥𝐴𝐵) ∪ {⟨𝐶, 𝐷⟩}))
bj-fvmptunsn.nel (𝜑 → ¬ 𝐶𝐴)
bj-fvmptunsn1.ex1 (𝜑𝐶𝑉)
bj-fvmptunsn1.ex2 (𝜑𝐷𝑊)
Assertion
Ref Expression
bj-fvmptunsn1 (𝜑 → (𝐹𝐶) = 𝐷)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝐷(𝑥)   𝐹(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem bj-fvmptunsn1
StepHypRef Expression
1 bj-fvmptunsn.un . 2 (𝜑𝐹 = ((𝑥𝐴𝐵) ∪ {⟨𝐶, 𝐷⟩}))
2 bj-fvmptunsn.nel . . 3 (𝜑 → ¬ 𝐶𝐴)
3 eqid 2735 . . . . 5 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
43dmmptss 6263 . . . 4 dom (𝑥𝐴𝐵) ⊆ 𝐴
54sseli 3991 . . 3 (𝐶 ∈ dom (𝑥𝐴𝐵) → 𝐶𝐴)
62, 5nsyl 140 . 2 (𝜑 → ¬ 𝐶 ∈ dom (𝑥𝐴𝐵))
7 bj-fvmptunsn1.ex1 . 2 (𝜑𝐶𝑉)
8 bj-fvmptunsn1.ex2 . 2 (𝜑𝐷𝑊)
91, 6, 7, 8bj-fununsn2 37237 1 (𝜑 → (𝐹𝐶) = 𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1537  wcel 2106  cun 3961  {csn 4631  cop 4637  cmpt 5231  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-nfc 2890  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-mpt 5232  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-iomnnom  37242
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