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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 6075 | . . . . 5 ⊢ (𝐹 = (𝐺 ∪ 𝐻) → (𝐹 “ {𝐴}) = ((𝐺 ∪ 𝐻) “ {𝐴})) | |
3 | imaundir 6173 | . . . . 5 ⊢ ((𝐺 ∪ 𝐻) “ {𝐴}) = ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴})) | |
4 | 2, 3 | eqtrdi 2791 | . . . 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 6124 | . . . . 5 ⊢ (¬ 𝐴 ∈ dom 𝐻 → (𝐻 “ {𝐴}) = ∅) | |
8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐻 “ {𝐴}) = ∅) |
9 | uneq2 4172 | . . . . 5 ⊢ ((𝐻 “ {𝐴}) = ∅ → ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴})) = ((𝐺 “ {𝐴}) ∪ ∅)) | |
10 | un0 4400 | . . . . 5 ⊢ ((𝐺 “ {𝐴}) ∪ ∅) = (𝐺 “ {𝐴}) | |
11 | 9, 10 | eqtrdi 2791 | . . . 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 2775 | . 2 ⊢ (𝜑 → (𝐹 “ {𝐴}) = (𝐺 “ {𝐴})) |
14 | bj-imafv 37234 | . 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 1537 ∈ wcel 2106 ∪ cun 3961 ∅c0 4339 {csn 4631 dom cdm 5689 “ cima 5692 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fv 6571 |
This theorem is referenced by: bj-fununsn1 37236 bj-fununsn2 37237 |
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