![]() |
Mathbox for BJ |
< Previous
Next >
Nearby theorems |
|
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 4148 | . . . 4 ⊢ (𝐺 ∪ {⟨𝐵, 𝐶⟩}) = ({⟨𝐵, 𝐶⟩} ∪ 𝐺) | |
3 | 1, 2 | eqtrdi 2782 | . . 3 ⊢ (𝜑 → 𝐹 = ({⟨𝐵, 𝐶⟩} ∪ 𝐺)) |
4 | bj-fununsn2.neldm | . . 3 ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐺) | |
5 | 3, 4 | bj-funun 36640 | . 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 583 | . 2 ⊢ (𝜑 → ({⟨𝐵, 𝐶⟩}‘𝐵) = 𝐶) |
10 | 5, 9 | eqtrd 2766 | 1 ⊢ (𝜑 → (𝐹‘𝐵) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ∈ wcel 2098 ∪ cun 3941 {csn 4623 ⟨cop 4629 dom cdm 5669 ‘cfv 6537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fv 6545 |
This theorem is referenced by: bj-fvsnun2 36644 bj-fvmptunsn1 36645 |
Copyright terms: Public domain | W3C validator |