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Theorem bj-funun 34662
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 5895 . . . . 5 (𝐹 = (𝐺𝐻) → (𝐹 “ {𝐴}) = ((𝐺𝐻) “ {𝐴}))
3 imaundir 5980 . . . . 5 ((𝐺𝐻) “ {𝐴}) = ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴}))
42, 3eqtrdi 2852 . . . 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 5937 . . . . 5 𝐴 ∈ dom 𝐻 → (𝐻 “ {𝐴}) = ∅)
86, 7syl 17 . . . 4 (𝜑 → (𝐻 “ {𝐴}) = ∅)
9 uneq2 4087 . . . . 5 ((𝐻 “ {𝐴}) = ∅ → ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴})) = ((𝐺 “ {𝐴}) ∪ ∅))
10 un0 4301 . . . . 5 ((𝐺 “ {𝐴}) ∪ ∅) = (𝐺 “ {𝐴})
119, 10eqtrdi 2852 . . . 4 ((𝐻 “ {𝐴}) = ∅ → ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴})) = (𝐺 “ {𝐴}))
128, 11syl 17 . . 3 (𝜑 → ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴})) = (𝐺 “ {𝐴}))
135, 12eqtrd 2836 . 2 (𝜑 → (𝐹 “ {𝐴}) = (𝐺 “ {𝐴}))
14 bj-imafv 34661 . 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 1538  wcel 2112  cun 3882  c0 4246  {csn 4528  dom cdm 5523  cima 5526  cfv 6328
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-xp 5529  df-cnv 5531  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fv 6336
This theorem is referenced by:  bj-fununsn1  34663  bj-fununsn2  34664
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