Theorem ovtpos

Description: The transposition swaps the arguments in a two-argument function. When F is a matrix, which is to say a function from ( 1 ... m ) X. ( 1 ... n ) to RR or some ring, tpos F is the transposition of F , which is where the name comes from. (Contributed by Mario Carneiro, 10-Sep-2015)

	Ref	Expression
Assertion	ovtpos	⊢ ( 𝐴 tpos 𝐹 𝐵 ) = ( 𝐵 𝐹 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		brtpos	⊢ ( 𝑦 ∈ V → ( ⟨ 𝐴 , 𝐵 ⟩ tpos 𝐹 𝑦 ↔ ⟨ 𝐵 , 𝐴 ⟩ 𝐹 𝑦 ) )
2	1	elv	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ tpos 𝐹 𝑦 ↔ ⟨ 𝐵 , 𝐴 ⟩ 𝐹 𝑦 )
3	2	iotabii	⊢ ( ℩ 𝑦 ⟨ 𝐴 , 𝐵 ⟩ tpos 𝐹 𝑦 ) = ( ℩ 𝑦 ⟨ 𝐵 , 𝐴 ⟩ 𝐹 𝑦 )
4		df-fv	⊢ ( tpos 𝐹 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ( ℩ 𝑦 ⟨ 𝐴 , 𝐵 ⟩ tpos 𝐹 𝑦 )
5		df-fv	⊢ ( 𝐹 ‘ ⟨ 𝐵 , 𝐴 ⟩ ) = ( ℩ 𝑦 ⟨ 𝐵 , 𝐴 ⟩ 𝐹 𝑦 )
6	3 4 5	3eqtr4i	⊢ ( tpos 𝐹 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ( 𝐹 ‘ ⟨ 𝐵 , 𝐴 ⟩ )
7		df-ov	⊢ ( 𝐴 tpos 𝐹 𝐵 ) = ( tpos 𝐹 ‘ ⟨ 𝐴 , 𝐵 ⟩ )
8		df-ov	⊢ ( 𝐵 𝐹 𝐴 ) = ( 𝐹 ‘ ⟨ 𝐵 , 𝐴 ⟩ )
9	6 7 8	3eqtr4i	⊢ ( 𝐴 tpos 𝐹 𝐵 ) = ( 𝐵 𝐹 𝐴 )