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Theorem bj-fvmptunsn1 37258
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 2737 . . . . 5 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
43dmmptss 6261 . . . 4 dom (𝑥𝐴𝐵) ⊆ 𝐴
54sseli 3979 . . 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 37255 1 (𝜑 → (𝐹𝐶) = 𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1540  wcel 2108  cun 3949  {csn 4626  cop 4632  cmpt 5225  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-nfc 2892  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-mpt 5226  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-iomnnom  37260
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