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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-fununsn2 | Structured version Visualization version GIF version | ||
| Description: Value of a function expressed as a union of a function and a singleton on a couple (with disjoint domain) at the first component of that couple. (Contributed by BJ, 18-Mar-2023.) (Proof modification is discouraged.) | 
| Ref | Expression | 
|---|---|
| bj-fununsn.un | ⊢ (𝜑 → 𝐹 = (𝐺 ∪ {〈𝐵, 𝐶〉})) | 
| bj-fununsn2.neldm | ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐺) | 
| bj-fununsn2.ex1 | ⊢ (𝜑 → 𝐵 ∈ 𝑉) | 
| bj-fununsn2.ex2 | ⊢ (𝜑 → 𝐶 ∈ 𝑊) | 
| Ref | Expression | 
|---|---|
| bj-fununsn2 | ⊢ (𝜑 → (𝐹‘𝐵) = 𝐶) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | bj-fununsn.un | . . . 4 ⊢ (𝜑 → 𝐹 = (𝐺 ∪ {〈𝐵, 𝐶〉})) | |
| 2 | uncom 4158 | . . . 4 ⊢ (𝐺 ∪ {〈𝐵, 𝐶〉}) = ({〈𝐵, 𝐶〉} ∪ 𝐺) | |
| 3 | 1, 2 | eqtrdi 2793 | . . 3 ⊢ (𝜑 → 𝐹 = ({〈𝐵, 𝐶〉} ∪ 𝐺)) | 
| 4 | bj-fununsn2.neldm | . . 3 ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐺) | |
| 5 | 3, 4 | bj-funun 37253 | . 2 ⊢ (𝜑 → (𝐹‘𝐵) = ({〈𝐵, 𝐶〉}‘𝐵)) | 
| 6 | bj-fununsn2.ex1 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
| 7 | bj-fununsn2.ex2 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑊) | |
| 8 | fvsng 7200 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ({〈𝐵, 𝐶〉}‘𝐵) = 𝐶) | |
| 9 | 6, 7, 8 | syl2anc 584 | . 2 ⊢ (𝜑 → ({〈𝐵, 𝐶〉}‘𝐵) = 𝐶) | 
| 10 | 5, 9 | eqtrd 2777 | 1 ⊢ (𝜑 → (𝐹‘𝐵) = 𝐶) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2108 ∪ cun 3949 {csn 4626 〈cop 4632 dom cdm 5685 ‘cfv 6561 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fv 6569 | 
| This theorem is referenced by: bj-fvsnun2 37257 bj-fvmptunsn1 37258 | 
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