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Theorem bj-fununsn2 34428
Description: Value of a function expressed as a union of a function 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-fununsn.un (𝜑𝐹 = (𝐺 ∪ {⟨𝐵, 𝐶⟩}))
bj-fununsn2.neldm (𝜑 → ¬ 𝐵 ∈ dom 𝐺)
bj-fununsn2.ex1 (𝜑𝐵𝑉)
bj-fununsn2.ex2 (𝜑𝐶𝑊)
Assertion
Ref Expression
bj-fununsn2 (𝜑 → (𝐹𝐵) = 𝐶)

Proof of Theorem bj-fununsn2
StepHypRef Expression
1 bj-fununsn.un . . . 4 (𝜑𝐹 = (𝐺 ∪ {⟨𝐵, 𝐶⟩}))
2 uncom 4126 . . . 4 (𝐺 ∪ {⟨𝐵, 𝐶⟩}) = ({⟨𝐵, 𝐶⟩} ∪ 𝐺)
31, 2syl6eq 2869 . . 3 (𝜑𝐹 = ({⟨𝐵, 𝐶⟩} ∪ 𝐺))
4 bj-fununsn2.neldm . . 3 (𝜑 → ¬ 𝐵 ∈ dom 𝐺)
53, 4bj-funun 34426 . 2 (𝜑 → (𝐹𝐵) = ({⟨𝐵, 𝐶⟩}‘𝐵))
6 bj-fununsn2.ex1 . . 3 (𝜑𝐵𝑉)
7 bj-fununsn2.ex2 . . 3 (𝜑𝐶𝑊)
8 fvsng 6934 . . 3 ((𝐵𝑉𝐶𝑊) → ({⟨𝐵, 𝐶⟩}‘𝐵) = 𝐶)
96, 7, 8syl2anc 584 . 2 (𝜑 → ({⟨𝐵, 𝐶⟩}‘𝐵) = 𝐶)
105, 9eqtrd 2853 1 (𝜑 → (𝐹𝐵) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1528  wcel 2105  cun 3931  {csn 4557  cop 4563  dom cdm 5548  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fv 6356
This theorem is referenced by:  bj-fvsnun2  34430  bj-fvmptunsn1  34431
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