Theorem tposfo

Description: The domain and codomain/range of a transposition. (Contributed by NM, 10-Sep-2015)

	Ref	Expression
Assertion	tposfo	⊢ ( 𝐹 : ( 𝐴 × 𝐵 ) –onto→ 𝐶 → tpos 𝐹 : ( 𝐵 × 𝐴 ) –onto→ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		relxp	⊢ Rel ( 𝐴 × 𝐵 )
2		tposfo2	⊢ ( Rel ( 𝐴 × 𝐵 ) → ( 𝐹 : ( 𝐴 × 𝐵 ) –onto→ 𝐶 → tpos 𝐹 : ◡ ( 𝐴 × 𝐵 ) –onto→ 𝐶 ) )
3	1 2	ax-mp	⊢ ( 𝐹 : ( 𝐴 × 𝐵 ) –onto→ 𝐶 → tpos 𝐹 : ◡ ( 𝐴 × 𝐵 ) –onto→ 𝐶 )
4		cnvxp	⊢ ◡ ( 𝐴 × 𝐵 ) = ( 𝐵 × 𝐴 )
5		foeq2	⊢ ( ◡ ( 𝐴 × 𝐵 ) = ( 𝐵 × 𝐴 ) → ( tpos 𝐹 : ◡ ( 𝐴 × 𝐵 ) –onto→ 𝐶 ↔ tpos 𝐹 : ( 𝐵 × 𝐴 ) –onto→ 𝐶 ) )
6	4 5	ax-mp	⊢ ( tpos 𝐹 : ◡ ( 𝐴 × 𝐵 ) –onto→ 𝐶 ↔ tpos 𝐹 : ( 𝐵 × 𝐴 ) –onto→ 𝐶 )
7	3 6	sylib	⊢ ( 𝐹 : ( 𝐴 × 𝐵 ) –onto→ 𝐶 → tpos 𝐹 : ( 𝐵 × 𝐴 ) –onto→ 𝐶 )