|   | Mathbox for BJ | < Previous  
      Next > Nearby theorems | |
| 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 6072 | . . . . 5 ⊢ (𝐹 = (𝐺 ∪ 𝐻) → (𝐹 “ {𝐴}) = ((𝐺 ∪ 𝐻) “ {𝐴})) | |
| 3 | imaundir 6169 | . . . . 5 ⊢ ((𝐺 ∪ 𝐻) “ {𝐴}) = ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴})) | |
| 4 | 2, 3 | eqtrdi 2792 | . . . 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 6120 | . . . . 5 ⊢ (¬ 𝐴 ∈ dom 𝐻 → (𝐻 “ {𝐴}) = ∅) | |
| 8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐻 “ {𝐴}) = ∅) | 
| 9 | uneq2 4161 | . . . . 5 ⊢ ((𝐻 “ {𝐴}) = ∅ → ((𝐺 “ {𝐴}) ∪ (𝐻 “ {𝐴})) = ((𝐺 “ {𝐴}) ∪ ∅)) | |
| 10 | un0 4393 | . . . . 5 ⊢ ((𝐺 “ {𝐴}) ∪ ∅) = (𝐺 “ {𝐴}) | |
| 11 | 9, 10 | eqtrdi 2792 | . . . 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 2776 | . 2 ⊢ (𝜑 → (𝐹 “ {𝐴}) = (𝐺 “ {𝐴})) | 
| 14 | bj-imafv 37253 | . 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 1539 ∈ wcel 2107 ∪ cun 3948 ∅c0 4332 {csn 4625 dom cdm 5684 “ cima 5687 ‘cfv 6560 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-xp 5690 df-cnv 5692 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fv 6568 | 
| This theorem is referenced by: bj-fununsn1 37255 bj-fununsn2 37256 | 
| Copyright terms: Public domain | W3C validator |