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Mirrors > Home > MPE Home > Th. List > df-tpos | Structured version Visualization version GIF version |
Description: Define the transposition of a function, which is a function 𝐺 = tpos 𝐹 satisfying 𝐺(𝑥, 𝑦) = 𝐹(𝑦, 𝑥). (Contributed by Mario Carneiro, 10-Sep-2015.) |
Ref | Expression |
---|---|
df-tpos | ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cF | . . 3 class 𝐹 | |
2 | 1 | ctpos 8050 | . 2 class tpos 𝐹 |
3 | vx | . . . 4 setvar 𝑥 | |
4 | 1 | cdm 5590 | . . . . . 6 class dom 𝐹 |
5 | 4 | ccnv 5589 | . . . . 5 class ◡dom 𝐹 |
6 | c0 4257 | . . . . . 6 class ∅ | |
7 | 6 | csn 4562 | . . . . 5 class {∅} |
8 | 5, 7 | cun 3886 | . . . 4 class (◡dom 𝐹 ∪ {∅}) |
9 | 3 | cv 1538 | . . . . . . 7 class 𝑥 |
10 | 9 | csn 4562 | . . . . . 6 class {𝑥} |
11 | 10 | ccnv 5589 | . . . . 5 class ◡{𝑥} |
12 | 11 | cuni 4840 | . . . 4 class ∪ ◡{𝑥} |
13 | 3, 8, 12 | cmpt 5158 | . . 3 class (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) |
14 | 1, 13 | ccom 5594 | . 2 class (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) |
15 | 2, 14 | wceq 1539 | 1 wff tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) |
Colors of variables: wff setvar class |
This definition is referenced by: tposss 8052 tposssxp 8055 brtpos2 8057 tposfun 8067 dftpos2 8068 dftpos4 8070 |
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