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Theorem bj-fvmptunsn1 34552
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 2820 . . . . 5 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
43dmmptss 6067 . . . 4 dom (𝑥𝐴𝐵) ⊆ 𝐴
54sseli 3938 . . 3 (𝐶 ∈ dom (𝑥𝐴𝐵) → 𝐶𝐴)
62, 5nsyl 142 . 2 (𝜑 → ¬ 𝐶 ∈ dom (𝑥𝐴𝐵))
7 bj-fvmptunsn1.ex1 . 2 (𝜑𝐶𝑉)
8 bj-fvmptunsn1.ex2 . 2 (𝜑𝐷𝑊)
91, 6, 7, 8bj-fununsn2 34549 1 (𝜑 → (𝐹𝐶) = 𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1537  wcel 2114  cun 3907  {csn 4539  cop 4545  cmpt 5118  dom cdm 5527  cfv 6327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5175  ax-nul 5182  ax-pow 5238  ax-pr 5302
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3472  df-sbc 3749  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4811  df-br 5039  df-opab 5101  df-mpt 5119  df-id 5432  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6286  df-fun 6329  df-fv 6335
This theorem is referenced by:  bj-iomnnom  34554
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