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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-funun | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
bj-funun.un | ⊢ (𝜑 → 𝐹 = (𝐺 ∪ 𝐻)) |
bj-funun.neldm | ⊢ (𝜑 → ¬ 𝐴 ∈ dom 𝐻) |
Ref | Expression |
---|---|
bj-funun | ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-funun.un | . . . 4 ⊢ (𝜑 → 𝐹 = (𝐺 ∪ 𝐻)) | |
2 | imaeq1 5715 | . . . . 5 ⊢ (𝐹 = (𝐺 ∪ 𝐻) → (𝐹 “ {𝐴}) = ((𝐺 ∪ 𝐻) “ {𝐴})) | |
3 | imaundir 5800 | . . . . 5 ⊢ ((𝐺 ∪ 𝐻) “ {𝐴}) = ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴})) | |
4 | 2, 3 | syl6eq 2830 | . . . 4 ⊢ (𝐹 = (𝐺 ∪ 𝐻) → (𝐹 “ {𝐴}) = ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴}))) |
5 | 1, 4 | syl 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}) = ∅) | |
8 | 6, 7 | syl 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}) | |
11 | 9, 10 | syl6eq 2830 | . . . 4 ⊢ ((𝐻 “ {𝐴}) = ∅ → ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴})) = (𝐺 “ {𝐴})) |
12 | 8, 11 | syl 17 | . . 3 ⊢ (𝜑 → ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴})) = (𝐺 “ {𝐴})) |
13 | 5, 12 | eqtrd 2814 | . 2 ⊢ (𝜑 → (𝐹 “ {𝐴}) = (𝐺 “ {𝐴})) |
14 | bj-imafv 33728 | . 2 ⊢ ((𝐹 “ {𝐴}) = (𝐺 “ {𝐴}) → (𝐹‘𝐴) = (𝐺‘𝐴)) | |
15 | 13, 14 | syl 17 | 1 ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐴)) |
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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