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Theorem bj-fvmptunsn1 36672
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 2727 . . . . 5 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
43dmmptss 6239 . . . 4 dom (𝑥𝐴𝐵) ⊆ 𝐴
54sseli 3974 . . 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 36669 1 (𝜑 → (𝐹𝐶) = 𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1534  wcel 2099  cun 3942  {csn 4624  cop 4630  cmpt 5225  dom cdm 5672  cfv 6542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fv 6550
This theorem is referenced by:  bj-iomnnom  36674
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