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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 6048 | . . . . 5 ⊢ (𝐹 = (𝐺 ∪ 𝐻) → (𝐹 “ {𝐴}) = ((𝐺 ∪ 𝐻) “ {𝐴})) | |
3 | imaundir 6144 | . . . . 5 ⊢ ((𝐺 ∪ 𝐻) “ {𝐴}) = ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴})) | |
4 | 2, 3 | eqtrdi 2782 | . . . 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 6096 | . . . . 5 ⊢ (¬ 𝐴 ∈ dom 𝐻 → (𝐻 “ {𝐴}) = ∅) | |
8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐻 “ {𝐴}) = ∅) |
9 | uneq2 4152 | . . . . 5 ⊢ ((𝐻 “ {𝐴}) = ∅ → ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴})) = ((𝐺 “ {𝐴}) ∪ ∅)) | |
10 | un0 4385 | . . . . 5 ⊢ ((𝐺 “ {𝐴}) ∪ ∅) = (𝐺 “ {𝐴}) | |
11 | 9, 10 | eqtrdi 2782 | . . . 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 2766 | . 2 ⊢ (𝜑 → (𝐹 “ {𝐴}) = (𝐺 “ {𝐴})) |
14 | bj-imafv 36639 | . 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 1533 ∈ wcel 2098 ∪ cun 3941 ∅c0 4317 {csn 4623 dom cdm 5669 “ cima 5672 ‘cfv 6537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-xp 5675 df-cnv 5677 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fv 6545 |
This theorem is referenced by: bj-fununsn1 36641 bj-fununsn2 36642 |
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