Definition df-tpos 8208

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.)

Assertion

Ref	Expression
df-tpos	⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))

Distinct variable group:   𝑥,𝐹

Detailed syntax breakdown of Definition df-tpos

Step	Hyp	Ref	Expression
1		cF	. . 3 class 𝐹
2	1	ctpos 8207	. 2 class tpos 𝐹
3		vx	. . . 4 setvar 𝑥
4	1	cdm 5641	. . . . . 6 class dom 𝐹
5	4	ccnv 5640	. . . . 5 class ◡dom 𝐹
6		c0 4299	. . . . . 6 class ∅
7	6	csn 4592	. . . . 5 class {∅}
8	5, 7	cun 3915	. . . 4 class (◡dom 𝐹 ∪ {∅})
9	3	cv 1539	. . . . . . 7 class 𝑥
10	9	csn 4592	. . . . . 6 class {𝑥}
11	10	ccnv 5640	. . . . 5 class ◡{𝑥}
12	11	cuni 4874	. . . 4 class ∪ ◡{𝑥}
13	3, 8, 12	cmpt 5191	. . 3 class (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})
14	1, 13	ccom 5645	. 2 class (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
15	2, 14	wceq 1540	1 wff tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))

This definition is referenced by:  tposss  8209  tposssxp  8212  brtpos2  8214  tposfun  8224  dftpos2  8225  dftpos4  8227  dftpos5  48866  dmtposss  48868  tposrescnv  48871