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Theorem bj-funun 33729
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 5715 . . . . 5 (𝐹 = (𝐺𝐻) → (𝐹 “ {𝐴}) = ((𝐺𝐻) “ {𝐴}))
3 imaundir 5800 . . . . 5 ((𝐺𝐻) “ {𝐴}) = ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴}))
42, 3syl6eq 2830 . . . 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 5756 . . . . 5 𝐴 ∈ dom 𝐻 → (𝐻 “ {𝐴}) = ∅)
86, 7syl 17 . . . 4 (𝜑 → (𝐻 “ {𝐴}) = ∅)
9 uneq2 3984 . . . . 5 ((𝐻 “ {𝐴}) = ∅ → ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴})) = ((𝐺 “ {𝐴}) ∪ ∅))
10 un0 4193 . . . . 5 ((𝐺 “ {𝐴}) ∪ ∅) = (𝐺 “ {𝐴})
119, 10syl6eq 2830 . . . 4 ((𝐻 “ {𝐴}) = ∅ → ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴})) = (𝐺 “ {𝐴}))
128, 11syl 17 . . 3 (𝜑 → ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴})) = (𝐺 “ {𝐴}))
135, 12eqtrd 2814 . 2 (𝜑 → (𝐹 “ {𝐴}) = (𝐺 “ {𝐴}))
14 bj-imafv 33728 . 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 1601  wcel 2107  cun 3790  c0 4141  {csn 4398  dom cdm 5355  cima 5358  cfv 6135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-xp 5361  df-cnv 5363  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fv 6143
This theorem is referenced by:  bj-fununsn1  33730  bj-fununsn2  33731
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