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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 4152 | . . . 4 ⊢ (𝐺 ∪ {〈𝐵, 𝐶〉}) = ({〈𝐵, 𝐶〉} ∪ 𝐺) | |
3 | 1, 2 | eqtrdi 2788 | . . 3 ⊢ (𝜑 → 𝐹 = ({〈𝐵, 𝐶〉} ∪ 𝐺)) |
4 | bj-fununsn2.neldm | . . 3 ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐺) | |
5 | 3, 4 | bj-funun 36121 | . 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 7174 | . . 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 2772 | 1 ⊢ (𝜑 → (𝐹‘𝐵) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1541 ∈ wcel 2106 ∪ cun 3945 {csn 4627 〈cop 4633 dom cdm 5675 ‘cfv 6540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fv 6548 |
This theorem is referenced by: bj-fvsnun2 36125 bj-fvmptunsn1 36126 |
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