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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 4168 | . . . 4 ⊢ (𝐺 ∪ {〈𝐵, 𝐶〉}) = ({〈𝐵, 𝐶〉} ∪ 𝐺) | |
3 | 1, 2 | eqtrdi 2791 | . . 3 ⊢ (𝜑 → 𝐹 = ({〈𝐵, 𝐶〉} ∪ 𝐺)) |
4 | bj-fununsn2.neldm | . . 3 ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐺) | |
5 | 3, 4 | bj-funun 37235 | . 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 2775 | 1 ⊢ (𝜑 → (𝐹‘𝐵) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1537 ∈ wcel 2106 ∪ cun 3961 {csn 4631 〈cop 4637 dom cdm 5689 ‘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-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fv 6571 |
This theorem is referenced by: bj-fvsnun2 37239 bj-fvmptunsn1 37240 |
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