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Theorem bj-funun 37254
Description: Value of a function expressed as a union of two functions at a point not in the domain of one of them. (Contributed by BJ, 18-Mar-2023.)
Hypotheses
Ref Expression
bj-funun.un (𝜑𝐹 = (𝐺𝐻))
bj-funun.neldm (𝜑 → ¬ 𝐴 ∈ dom 𝐻)
Assertion
Ref Expression
bj-funun (𝜑 → (𝐹𝐴) = (𝐺𝐴))

Proof of Theorem bj-funun
StepHypRef Expression
1 bj-funun.un . . . 4 (𝜑𝐹 = (𝐺𝐻))
2 imaeq1 6072 . . . . 5 (𝐹 = (𝐺𝐻) → (𝐹 “ {𝐴}) = ((𝐺𝐻) “ {𝐴}))
3 imaundir 6169 . . . . 5 ((𝐺𝐻) “ {𝐴}) = ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴}))
42, 3eqtrdi 2792 . . . 4 (𝐹 = (𝐺𝐻) → (𝐹 “ {𝐴}) = ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴})))
51, 4syl 17 . . 3 (𝜑 → (𝐹 “ {𝐴}) = ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴})))
6 bj-funun.neldm . . . . 5 (𝜑 → ¬ 𝐴 ∈ dom 𝐻)
7 ndmima 6120 . . . . 5 𝐴 ∈ dom 𝐻 → (𝐻 “ {𝐴}) = ∅)
86, 7syl 17 . . . 4 (𝜑 → (𝐻 “ {𝐴}) = ∅)
9 uneq2 4161 . . . . 5 ((𝐻 “ {𝐴}) = ∅ → ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴})) = ((𝐺 “ {𝐴}) ∪ ∅))
10 un0 4393 . . . . 5 ((𝐺 “ {𝐴}) ∪ ∅) = (𝐺 “ {𝐴})
119, 10eqtrdi 2792 . . . 4 ((𝐻 “ {𝐴}) = ∅ → ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴})) = (𝐺 “ {𝐴}))
128, 11syl 17 . . 3 (𝜑 → ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴})) = (𝐺 “ {𝐴}))
135, 12eqtrd 2776 . 2 (𝜑 → (𝐹 “ {𝐴}) = (𝐺 “ {𝐴}))
14 bj-imafv 37253 . 2 ((𝐹 “ {𝐴}) = (𝐺 “ {𝐴}) → (𝐹𝐴) = (𝐺𝐴))
1513, 14syl 17 1 (𝜑 → (𝐹𝐴) = (𝐺𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1539  wcel 2107  cun 3948  c0 4332  {csn 4625  dom cdm 5684  cima 5687  cfv 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-xp 5690  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fv 6568
This theorem is referenced by:  bj-fununsn1  37255  bj-fununsn2  37256
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