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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 5909 | . . . . 5 ⊢ (𝐹 = (𝐺 ∪ 𝐻) → (𝐹 “ {𝐴}) = ((𝐺 ∪ 𝐻) “ {𝐴})) | |
3 | imaundir 5994 | . . . . 5 ⊢ ((𝐺 ∪ 𝐻) “ {𝐴}) = ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴})) | |
4 | 2, 3 | eqtrdi 2787 | . . . 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 5951 | . . . . 5 ⊢ (¬ 𝐴 ∈ dom 𝐻 → (𝐻 “ {𝐴}) = ∅) | |
8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐻 “ {𝐴}) = ∅) |
9 | uneq2 4057 | . . . . 5 ⊢ ((𝐻 “ {𝐴}) = ∅ → ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴})) = ((𝐺 “ {𝐴}) ∪ ∅)) | |
10 | un0 4291 | . . . . 5 ⊢ ((𝐺 “ {𝐴}) ∪ ∅) = (𝐺 “ {𝐴}) | |
11 | 9, 10 | eqtrdi 2787 | . . . 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 2771 | . 2 ⊢ (𝜑 → (𝐹 “ {𝐴}) = (𝐺 “ {𝐴})) |
14 | bj-imafv 35106 | . 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 1543 ∈ wcel 2112 ∪ cun 3851 ∅c0 4223 {csn 4527 dom cdm 5536 “ cima 5539 ‘cfv 6358 |
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 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-xp 5542 df-cnv 5544 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fv 6366 |
This theorem is referenced by: bj-fununsn1 35108 bj-fununsn2 35109 |
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