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Theorem bj-funun 36121
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 6052 . . . . 5 (𝐹 = (𝐺𝐻) → (𝐹 “ {𝐴}) = ((𝐺𝐻) “ {𝐴}))
3 imaundir 6147 . . . . 5 ((𝐺𝐻) “ {𝐴}) = ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴}))
42, 3eqtrdi 2788 . . . 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 6099 . . . . 5 𝐴 ∈ dom 𝐻 → (𝐻 “ {𝐴}) = ∅)
86, 7syl 17 . . . 4 (𝜑 → (𝐻 “ {𝐴}) = ∅)
9 uneq2 4156 . . . . 5 ((𝐻 “ {𝐴}) = ∅ → ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴})) = ((𝐺 “ {𝐴}) ∪ ∅))
10 un0 4389 . . . . 5 ((𝐺 “ {𝐴}) ∪ ∅) = (𝐺 “ {𝐴})
119, 10eqtrdi 2788 . . . 4 ((𝐻 “ {𝐴}) = ∅ → ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴})) = (𝐺 “ {𝐴}))
128, 11syl 17 . . 3 (𝜑 → ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴})) = (𝐺 “ {𝐴}))
135, 12eqtrd 2772 . 2 (𝜑 → (𝐹 “ {𝐴}) = (𝐺 “ {𝐴}))
14 bj-imafv 36120 . 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 1541  wcel 2106  cun 3945  c0 4321  {csn 4627  dom cdm 5675  cima 5678  cfv 6540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-xp 5681  df-cnv 5683  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fv 6548
This theorem is referenced by:  bj-fununsn1  36122  bj-fununsn2  36123
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