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Theorem bj-funun 37625
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 6013 . . . . 5 (𝐹 = (𝐺𝐻) → (𝐹 “ {𝐴}) = ((𝐺𝐻) “ {𝐴}))
3 imaundir 6104 . . . . 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 6061 . . . . 5 𝐴 ∈ dom 𝐻 → (𝐻 “ {𝐴}) = ∅)
86, 7syl 17 . . . 4 (𝜑 → (𝐻 “ {𝐴}) = ∅)
9 uneq2 4094 . . . . 5 ((𝐻 “ {𝐴}) = ∅ → ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴})) = ((𝐺 “ {𝐴}) ∪ ∅))
10 un0 4324 . . . . 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 37624 . 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 1548  wcel 2121  cun 3882  c0 4263  {csn 4557  dom cdm 5620  cima 5623  cfv 6488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5220  ax-nul 5230  ax-pr 5364
This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-opab 5137  df-xp 5626  df-cnv 5628  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6444  df-fv 6496
This theorem is referenced by:  bj-fununsn1  37626  bj-fununsn2  37627
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